# On a continuation of quaternionic and octonionic logarithm along curves and
the winding number

Graziano Gentili  DiMaI, Università di Firenze, Viale Morgagni 67/A, Firenze,
Italy<EMAIL_ADDRESS>,  Jasna Prezelj  Fakulteta za matematiko in
fiziko Jadranska 21 1000 Ljubljana, Slovenija, UP FAMNIT, Glagoljaška 8, Koper
Slovenija, IMFM, Jadranska 19, Slovenia<EMAIL_ADDRESS>and
Fabio Vlacci  DiSPeS Università di Trieste Piazzale Europa 1, Trieste, Italy
<EMAIL_ADDRESS>
The first and third authors were partly supported by INdAM, through: GNSAGA;
INdAM project “Hypercomplex function theory and applications”. It was also
partly supported by MIUR, through the projects: Finanziamento Premiale FOE
2014 “Splines for accUrate NumeRics: adaptIve models for Simulation
Environments”. The second author was partially supported by research program
P1-0291 and by research project J1-3005 at Slovenian Research Agency.

##  1\. Introduction

This paper focuses on the problem of finding a continuous extension of the
hypercomplex logarithm along a path. As pointed out in [GPV], while a branch
of the complex logarithm can be defined in a small open neighborhood of a
strictly negative real point, no continuous branch of the hypercomplex
logarithm can be defined in any open set $A\subset\mathbb{K}\setminus\\{0\\}$
which contains a strictly negative real point $x_{0}$ (here $\mathbb{K}$
represents the algebra of quaternions or of octonions).

To overcome these difficulties, in [GPV] we introduced the logarithmic
manifold $\mathscr{E}_{\mathbb{K}}^{+}$ and then showed that, if
$q\in\mathbb{K},\ q=x+Iy$ then $E(x+Iy)=(\exp x\cos y+I\exp x\sin y,Iy)$ is an
immersion and a diffeomorphism between $\mathbb{K}$ and
$\mathscr{E}_{\mathbb{K}}^{+}$.

In this paper, we consider lifts of paths in $\mathbb{K}\setminus\\{0\\}$ to
the logarithmic manifold $\mathscr{E}^{+}_{\mathbb{K}}$; even though
$\mathbb{K}\setminus\\{0\\}$ is simply connected, in general, given a path in
$\mathbb{K}\setminus\\{0\\}$, the existence of a lift of this path to
$\mathscr{E}^{+}_{\mathbb{K}}$ is not guaranteed. There is an obvious
equivalence between the problem of lifting a path in
$\mathbb{K}\setminus\\{0\\}$ and the one of finding a continuation of the
hypercomplex logarithm $\log_{\mathbb{K}}$ along this path.

We want to recall that the slice regular logarithm $\log_{*}(f)$ of a slice
regular function $f$ (see [AdF1, GPV1]) over the quaternions or octonions,
introduced as the slice regular inverse of the slice regular exponential
$\exp_{*}(f)$ of a slice regular function $f$ (see [AdF]), is not defined in
general via the lift of $f$ to $\mathscr{E}_{\mathbb{K}}^{+}$. In particular
it turns out that, in general, $\log_{*}(f)(q)\neq\log_{\mathbb{K}}(f(q))$.

The paper is organized as follows: in Sections 2 and 3, after recalling the
basic notions on slice regular exponential and logarithmic functions, we
provide explicit examples of paths intersecting the real axis and show how a
branch of the hypercomplex logarithm can be defined along certain curves even
when they encounter the real axis at negative points, providing a so called
_continuation of the logarithm along a continuous curve_.

Furthermore, we introduce the notion of path and of loop with a companion (see
Subsection 4.1) and then give a definition of winding number with respect to
$0$ that has a full meaning for a class of loops in
$\mathbb{K}\setminus\\{0\\}\simeq\mathbb{R}^{2^{s}}\setminus\\{0\\}\ (s=2,3)$
with companion; this fact is quite novel and original since it is well known
that a definition of winding number for a loop (with respect to a point) is
not in general possible in $\mathbb{R}^{n}$ when $n$ is greater than $2$.
Moreover this notion of winding number is invariant for the class of
$c-$homotopic loops with companion.

Finally, in the last Section 5, we extend the continuation of the hypercomplex
logarithm to the case curves with an infinite number of intersections with the
real axis. These represent the set of obstructions for such an extension. When
these obstructions are “mild” and “reasonable”, then we also present an
effective way to calculate the winding numbers using the so-called notion of
signature.

##  2\. Preliminary results

We denote by $\mathbb{K}$ either the algebra of quaternions or octonions. Let
$\mathbb{S}$ be the sphere of imaginary units, i.e. the set of
$I\in\mathbb{K}$ such that $I^{2}=-1$. Given any
$z\in\mathbb{K}\setminus\mathbb{R},$ there exist (and are uniquely determined)
an imaginary unit $I$, and two real numbers $x,y$ (with $y>0$) such that
$z=x+Iy$. With this notation, the conjugate of $z$ will be $\bar{z}:=x-Iy$ and
$|z|^{2}=z\bar{z}=\bar{z}z=x^{2}+y^{2}$. Each imaginary unit $I$ generates (as
a real algebra) a copy of a complex plane denoted by $\mathbb{C}_{I}$. We call
such a complex plane a slice. The upper half-plane in $\mathbb{C}_{I}$, namely
$\\{x+yI\ :\ y>0\\}$ will be denoted by $\mathbb{C}_{I}^{+}$. Similarly, the
lower half-plane in $\mathbb{C}_{I}$ $\\{x+yI\ :\ y<0\\}$ will be denoted by
$\mathbb{C}_{I}^{-}$; each of these two half–planes will be called a leaf of
$\mathbb{C}_{I}$. On any leaf $\mathbb{C}_{I}^{+}$ we define the function
$\arg_{I}:\mathbb{C}_{I}^{+}\to(0,\pi)$ as
$z=x+Iy\in\mathbb{C}_{I}^{+}\mapsto\cot^{-1}(x/y):=\arg_{I}(z)$.

The function $\arg_{I}$ can be continuously extended as a function
$\arg_{I}:\mathbb{C}_{I}^{+}\cup\mathbb{R}^{+}\cup\mathbb{R}^{-}\to[0,\pi]$.

It is also useful to define the imaginary unit function on
$\mathbb{K}\setminus\mathbb{R}$ in the following way: if
$z\in\mathbb{C}_{I}^{+},$ i.e. if $z=x+Iy$, with $x,y\in\mathbb{R}$ and $y>0$,
then $\mathcal{I}(z)=I$; if $z\in\mathbb{C}_{I}^{-},$ i.e. if $z=x-Iy$, with
$x,y\in\mathbb{R}$ and $y>0$, then $\mathcal{I}(z)=-I$.

######  Remark 2.1.

_It is worthwhile noticing that the function $\mathcal{I}$ cannot be extended
as a continuous function to any single point of the real axis $\mathbb{R}$ of
$\mathbb{K}$. At the same time, if we set
$\mathbb{S}(-\pi,\pi)=\\{Iy:I\in\mathbb{S},\ y\in(-\pi,\pi)\\}$, then the
function_

$\textnormal{Arg}:\mathbb{K}\setminus(-\infty,0]\to\mathbb{S}(-\pi,\pi)$

_defined as the product_

$\textnormal{Arg}(q):=\mathcal{I}(q)\arg_{\mathcal{I}}(q)$

_can be extended (as the zero function) to the positive real axis
$\mathbb{R}^{+}$ of $\mathbb{K}$._

##  3\. The hypercomplex exponential and logarithm

Let us recall that the exponential map on $\mathbb{K}$

$\exp:\mathbb{K}\to\mathbb{K}\setminus\\{0\\}$

defined as

$\exp(q)=\sum_{k\geq 0}\dfrac{q^{k}}{k!}$

is a slice regular and slice preserving entire function on $\mathbb{K}$ ([AdF,
GSS]). Let $\mathscr{E}_{\mathbb{K}}^{+}=f(\mathbb{K}^{+})$ denote the
_logarithm manifold_ , i.e., the image $T(\mathbb{K}^{+})$ of
$\mathbb{K}^{+}=\\{q\in\mathbb{K}:{\rm Re}\ q>0\\}$ of the map
$T:\mathbb{K}\to\mathbb{K}\times{\rm Im}(\mathbb{K})$ defined by

$\displaystyle T(x+Iy)=(\sinh x\cos y+I\sinh x\sin y,Iy)$

for $I\in\mathbb{S}$, $x,y\in\mathbb{R}$. The $\mathscr{E}_{\mathbb{K}}^{+}$\-
exponential map

$E:\mathbb{K}\to\mathscr{E}_{\mathbb{K}}^{+}\subset\mathbb{K}\times{\rm
Im}(\mathbb{K})$

defined by:

$E(x+Iy)=(\exp(x+Iy),Iy)=(\exp x\cos y+I\exp x\sin y,Iy)$

is an immersion and a diffeomorphism between $\mathbb{K}$ and
$\mathscr{E}_{\mathbb{K}}^{+}$ (see [GPV]). In the case of quaternions, it
endows $\mathscr{E}_{\mathbb{H}}^{+}$ with a structure of slice quaternionic
manifold (see, e.g., [GGS]), which is different from the structure of
hypercomplex Riemann manifold defined in Propostion 4.3. [GPV])

The next definition and result appear in [GPV].

######  Definition 3.1.

Let $\mathscr{E}_{\mathbb{K}}^{+}$ be the semi-helicoidal hypercomplex
manifold. The $\mathscr{E}_{\mathbb{K}}^{+}$\- logarithm

$L:\mathscr{E}_{\mathbb{K}}^{+}\subset\mathbb{K}\times{\rm
Im}(\mathbb{K})\to\mathbb{K}$

is defined as follows, in terms of the real logarithm $\log$:

$L(q,p)=\log|q|+p$

Indeed, if $(q,p)\in\ \mathscr{E}_{\mathbb{K}}^{+}$, then $q=r\exp p$ for
$r=|q|$ and our definition can be rewritten as:

$L(r\exp p,p)=\log r+p$

The hypercomplex manifold $\mathscr{E}_{\mathbb{K}}^{+}$ plays the role of an
“adapted” blow-up of $\mathbb{K}$ at points of the form $x+2Ik\pi$, for
$k\in\mathbb{Z}$ and $k\neq 0$.

######  Proposition 3.2.

The map

$L:\mathscr{E}_{\mathbb{K}}^{+}\to\mathbb{K}$

is the inverse of the $\mathscr{E}_{\mathbb{K}}^{+}$\- exponential $E$, and a
diffeomorphism from the logarithm manifold $\mathscr{E}_{\mathbb{K}}^{+}$ to
$\mathbb{K}$.

Note that if ${\mathrm{pr}_{1}}:\mathbb{K}\times{\rm
Im}(\mathbb{K})\to\mathbb{K}$ denotes the projection on the first factor, then
by definition the following equality holds

${\mathrm{pr}_{1}}\circ E(q)=\exp(q)$

for all $q\in\mathbb{K}$. Indeed, the map $L$ is a slice regular map from
$\mathscr{E}^{+}$ to $\mathbb{H}$, with respect to the structure of slice
regular manifold induced by $E$ on $\mathscr{E}_{\mathbb{K}}^{+}$ (see, e.g.,
[GGS]). This map allows the definition of the hypercomplex logarithm (see
[GPV, GPV1]):

######  Definition 3.3.

Let ${\mathrm{pr}_{1}}:\mathscr{E}_{\mathbb{K}}^{+}\subset\mathbb{K}\times{\rm
Im}(\mathbb{K})\to\mathbb{K}\setminus\\{0\\}$ denote the natural projection

$(q,p)\mapsto q$

and let $\Omega\subset\mathscr{E}_{\mathbb{K}}^{+}$ be a path connected subset
such that ${\mathrm{pr}_{1}}_{|_{\Omega}}$ is injective. Then, the map

$\log_{\mathbb{K}}:{\mathrm{pr}_{1}}(\Omega)\to\mathbb{K}$

defined by

$\log_{\mathbb{K}}q=L({\mathrm{pr}_{1}}_{|_{\Omega}}^{-1}(q))$

is called a _branch_ or _a determination of the hypercomplex logarithm_ on
${\mathrm{pr}_{1}}(\Omega)$.

As one can expect, it holds

$\exp(\log_{\mathbb{K}}q)={\mathrm{pr}_{1}}(E(L({\mathrm{pr}_{1}}_{|_{\Omega}}^{-1}(q))))={\mathrm{pr}_{1}}({\mathrm{pr}_{1}}_{|_{\Omega}}^{-1}(q))=q$

for all $q$ in ${\mathrm{pr}_{1}}(\Omega)$. It is worthwhile noticing that, if
we consider the open, path-connected subset

$\Omega=\\{(q,\textnormal{Arg}(q)):q\in\mathbb{K}\setminus(-\infty,\pi]\\}\subset\mathscr{E}_{\mathbb{K}}^{+},$

then the projection on the first factor

${\mathrm{pr}_{1}}:\Omega\to\mathbb{K}\setminus(-\infty,0]$

is injective. Therefore, in this way, one defines the principal branch of the
logarithm (see [GV]) in
${\mathrm{pr}_{1}}(\Omega)=\mathbb{K}\setminus(-\infty,\pi]$ (see [GPV1,
AdF1]). The principal branch of the hypercomplex logarithm

$\log_{0}:\mathbb{K}\setminus(-\infty,0]\to\mathbb{R}\times[0,\pi)\mathbb{S}\subset\mathbb{K}$
$q\mapsto\log|q|+\textnormal{Arg}(q)$

is well defined and, for all $I\in\mathbb{S}$, coincides with the principal
branch of the complex logarithm in the slice
$\mathbb{C}_{I}\setminus(-\infty,0]$. As a consequence, $\log_{0}$ is a slice
regular function in the symmetric slice domain
$\mathbb{K}\setminus(-\infty,0]$.

As already observed in the Introduction, despite the analogy with the complex
holomorphic case, in general no continuous branch of the hypercomplex
logarithm can be defined in any open set $A\subset\mathbb{K}\setminus\\{0\\}$
which contains a strictly negative real point $x_{0}$. Nevertheless, we will
now see how a branch of the hypercomplex logarithm can be defined along
certain curves even when they encounter the real axis at negative points,
providing a so called _continuation of the logarithm along a continuous
curve_.

Throughout the paper, a continuous curve will be called a _path_ , and a
closed path will be called a _loop_.

######  Definition 3.4.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a path. Then a path
$\widetilde{\gamma}:[a,b]\rightarrow\mathbb{K}$ is called a continuation of
the logarithm along $\gamma$ if

$\exp\circ\widetilde{\gamma}=\gamma,$

i.e., if the following diagram commutes:

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathbb{K}}$$\scriptstyle{\exp}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces[a,b]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{\gamma}}$$\scriptstyle{\gamma}$$\textstyle{\mathbb{K}\setminus\\{0\\}}$

The point $\widetilde{\gamma}(a)\in\mathbb{K}$ will be called the initial
point of the continuation $\widetilde{\gamma}$.

To study the possible continuations of the logarithm along a path, we need to
specifically define the various branches of the hypercomplex argument of an
element from $\mathbb{K}\setminus\\{0\\}$.

######  Definition 3.5.

If $k\in\mathbb{Z}$, for all $q\in\mathbb{K}\setminus\mathbb{R}$,
$q=x+\mathcal{I}y$ with $y>0$, let us define for

$\begin{array}[]{ll}k=2l:&\mathcal{I}_{2l}(q)=\mathcal{I}(q),\,\,\arg_{2l}(q)=\arg_{\mathcal{I}}(q)+2l\pi,\\\
&\mbox{where }\arg_{2l}(q)\in(2l\pi,(2l+1)\pi),\\\ &\\\
k=2l+1:&\mathcal{I}_{2l+1}(q)=-\mathcal{I}(q),\,\,\arg_{2l+1}(q)=(2\pi-\arg_{\mathcal{I}}(q))+2l\pi,\\\
&\mbox{where }\arg_{2l+1}(q)\in((2l+1)\pi,(2l+2)\pi).\end{array}$

The $k$-th branch of the hypercomplex argument

$\textnormal{Arg}_{k}:\mathbb{K}\setminus\mathbb{R}\rightarrow\mathbb{S}(k\pi,(k+1)\pi)$

is defined by setting

$\textnormal{Arg}_{k}(q):=\mathcal{I}_{k}(q)\arg_{k}(q).$

As a consequence,

$\displaystyle\textnormal{Arg}_{2l+1}(q)$ $\displaystyle=$
$\displaystyle-\mathcal{I}(q)(2\pi-\arg_{\mathcal{I}}(q)+2l\pi)$
$\displaystyle=$
$\displaystyle\mathcal{I}(q)(\arg_{\mathcal{I}}(q)-2(l+1)\pi)$
$\displaystyle=$ $\displaystyle\mathcal{I}(q)\arg_{-2(l+1)}(q)$
$\displaystyle=$ $\displaystyle\textnormal{Arg}_{-(2l+2)}(q)$

Therefore, the only different branches of the hypercomplex argument of a
quaternion $q\in\mathbb{K}\setminus\mathbb{R}$, $q=x+\mathcal{I}y$ with $y>0$,
can be listed for $k\in\mathbb{Z}$ as

$\textnormal{Arg}_{2k}(q):=\mathcal{I}(q)\arg_{2k}(q).$

It is worthwhile noticing that for any fixed
$q\in\mathbb{K}\setminus\mathbb{R}$, we have that for all $k\in\mathbb{Z}$

$\exp(\textnormal{Arg}_{2k}(q))=\exp(\textnormal{Arg}(q));$

indeed

$\displaystyle\exp(\textnormal{Arg}_{2l}(q))$ $\displaystyle=$
$\displaystyle\exp[{\mathcal{I}}(q)(\arg_{\mathcal{I}}(q)+2l\pi)]=\exp[{{\mathcal{I}}(q)\arg_{\mathcal{I}}(q)}]$
$\displaystyle=$ $\displaystyle\exp(\textnormal{Arg}(q)).$

##  4\. Continuation of hypercomplex logarithms along paths

The construction of a continuation of the logarithm along a path naturally
involves the notion of a lift of a path.

######  Definition 4.1.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a path. Then a path
$\Gamma:[a,b]\rightarrow\mathcal{E}_{\mathbb{K}}^{+}$ is a lift of $\gamma$
(to $\mathcal{E}_{\mathbb{K}}^{+}$) if ${\mathrm{pr}_{1}}\circ\Gamma=\gamma$,
i.e., if the the following diagram commutes:

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{E}_{\mathbb{K}}^{+}}$$\scriptstyle{{\mathrm{pr}_{1}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces[a,b]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma}$$\scriptstyle{\gamma}$$\textstyle{\mathbb{K}\setminus\\{0\\}}$

For $(q,p)\in\mathcal{E}_{\mathbb{K}}^{+}$, a lift $\Gamma$ of $\gamma$ such
that $\Gamma(a)=(q,p)$ will be said to have _initial point_ $(q,p)$.

The existence of a lift of a path $\gamma$ is equivalent to the existence of a
continuation of the hypercomplex logarithm along it.

######  Proposition 4.2.

Let $\gamma:[a,b]\subset\mathbb{R}\rightarrow\mathbb{K}\setminus\\{0\\}$ be a
path. Then, there exists a lift of $\gamma$ to $\mathcal{E}_{\mathbb{K}}^{+}$
if, and only if, there exists a continuation of the logarithm along $\gamma$.

###### Proof.

Suppose that there exists a continuation of the logarithm $\widetilde{\gamma}$
along $\gamma$. Then the path $\Gamma$ defined by
$\Gamma(t)=((\exp\circ\widetilde{\gamma})(t),{\rm Im}(\widetilde{\gamma}(t))$
is obviously a lift of $\gamma$ to $\mathcal{E}_{\mathbb{K}}^{+}$.

|
---|---
$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{E}_{\mathbb{K}}^{+}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathrm{pr}_{1}}}$$\scriptstyle{L}$$\textstyle{\mathbb{K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces[a,b]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Gamma}$$\scriptstyle{\gamma}$$\textstyle{\mathbb{K}\setminus\\{0\\}}$

Conversely, if a lift $\Gamma$ of the path
$\gamma:[a,b]\to\mathbb{K}\setminus\\{0\\}$ to $\mathcal{E}^{+}_{\mathbb{K}}$
exists, then a continuation of the hypercomplex logarithm along $\gamma$ can
be defined by $\widetilde{\gamma}(t):=L(\Gamma(t))$. ∎

Thanks to the result just stated, we are left to find conditions under which a
path $\gamma:[a,b]\to\mathbb{K}\setminus\\{0\\}$ can be lifted to
$\mathcal{E}^{+}_{\mathbb{K}}$. Since the map
${\mathrm{pr}_{1}}:\mathcal{E}_{\mathbb{K}}^{+}\to\mathbb{K}\setminus\\{0\\}$
is not a covering, we have to specifically study the existence of lifts of
$\gamma$.

Let us first consider the easy cases: it is not difficult to see that if we
restrict the map
${\mathrm{pr}_{1}}:\mathcal{E}_{\mathbb{K}}^{+}\to\mathbb{K}\setminus\\{0\\}$
to the preimage of $\mathbb{K}\setminus\mathbb{R}$, then the restriction
${\mathrm{pr}_{1}}_{|{\mathrm{pr}_{1}}^{-1}(\mathbb{K}\setminus\mathbb{R})}$
becomes a covering. Indeed it becomes a trivial covering, since
${\mathrm{pr}_{1}}^{-1}(\mathbb{K}\setminus\mathbb{R})$ is homeomorphic
(namely diffeomorphic) through the diffeomorphism

$E:\mathbb{K}\to\mathscr{E}_{\mathbb{K}}^{+}$

to the countable collection of open simply connected domains given by

$\mathbb{K}\setminus\left\\{\bigcup_{k\in\mathbb{Z}}\mathbb{R}\times\mathbb{S}k\pi\right\\}=\bigcup_{k\in\mathbb{Z}}\mathbb{R}\times\mathbb{S}(k\pi,(k+1)\pi).$

Let us now set, for any $k\in\mathbb{Z}$,

$D_{k}=\mathbb{R}\times\mathbb{S}(k\pi,(k+1)\pi),\qquad
E(D_{k})=E(\mathbb{R}\times\mathbb{S}(k\pi,(k+1)\pi))\subset\mathscr{E}_{\mathbb{K}}^{+}.$

Notice that $\mathbb{S}(k\pi,(k+1)\pi)=\mathbb{S}(-(k+1)\pi,-k\pi))$ and hence
$D_{2k}=D_{-(2k+1)}.$ With this in mind, we can now state the following
proposition.

######  Proposition 4.3.

Assume the path $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ is such
that $\gamma([a,b])\cap\mathbb{R}=\varnothing$, and let

$\gamma(t)=x(t)+{\mathcal{I}}(t)y(t)$

with $y(t)>0$ for all $t\in[a,b]$. Then, for any $k\in\mathbb{Z}$, there
exists one, and only one, lift $\Gamma_{k}$ of $\gamma$ to
$E(D_{2k})\subset\mathscr{E}_{\mathbb{K}}^{+}$ with initial point

$\Gamma_{k}(a)=(\gamma(a),\textnormal{Arg}_{2k}(\gamma(a))).$

Namely, for all $t\in[a,b]$, we have

$\Gamma_{k}(t):=(\gamma(t),\textnormal{Arg}_{2k}(\gamma(t)))\in E(D_{2k}).$

Finally, for all $k\in\mathbb{Z}$, the map defined on the interval $[a,b]$ by

(4.2)
$(\log_{k}\circ\gamma)(t):=(L\circ\Gamma_{k})(t)=\log|\gamma(t)|+\textnormal{Arg}_{2k}(\gamma(t))$

is the unique continuation of the hypercomplex logarithm along $\gamma$ with
initial point $\log|\gamma(a)|+\textnormal{Arg}_{2k}(\gamma(a))$, and is
called _the $k$-th branch of the hypercomplex logarithm along $\gamma$ with
initial point $\log|\gamma(a)|+\textnormal{Arg}_{2k}(\gamma(a))$._

###### Proof.

For each $k\in\mathbb{Z}$, the proof of the existence and uniqueness of
$\Gamma_{k}$ as in the statement is a straightforward consequence of what
already established. To prove the last part of the statement, let us consider
the graph $\Omega_{k}$ of the lift $\Gamma_{k}$ of $\gamma$ to
$E(D_{2k})\subset\mathscr{E}_{\mathbb{K}}^{+}$ with initial point
$\Gamma_{k}(a)=(\gamma(a),\textnormal{Arg}_{2k}(\gamma(a)))$, i.e.,

$\Omega_{k}:=\\{(q,\textnormal{Arg}_{2k}(q)):q\in\gamma([a,b])\\}\subset
E(D_{2k})\subset\mathcal{E}^{+}_{\mathbb{K}}.$

Since the projection
$\pi:\mathcal{E}^{+}_{\mathbb{K}}\to\mathbb{K}\setminus\\{0\\}$ restricted to
$\Omega_{k}$ is injective, following Definition 3.3, we obtain (4.2). ∎

Under the hypotheses of the preceding proposition, loops lift to loops, hence:

######  Corollary 4.4.

Assume the loop $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ is such
that $\gamma([a,b])\cap\mathbb{R}=\varnothing$. Then for each
$k\in\mathbb{Z}$, the lift $\Gamma_{k}$ found in Proposition 4.3 is a loop. As
a consequence, for each $k\in\mathbb{Z}$,

$\log_{k}(\gamma(a))=\log_{k}(\gamma(b))$

Among the initial cases, there is the one corresponding to what is stated in
Remark 2.1.

######  Proposition 4.5.

Assume the path $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ is such
that $\gamma([a,b])\cap\mathbb{R}^{-}=\varnothing$. Then there exists a lift
$\Gamma$ of $\gamma$ to $\mathcal{E}^{+}_{\mathbb{K}}$.

###### Proof.

The proof is a consequence of what observed in Remark 2.1. Indeed, taking into
account that $\mathbb{S}(-\pi,\pi)=\mathbb{S}[0,\pi)=\mathbb{S}(-\pi,0]$, the
mentioned remark implies that
$\pi:E(\mathbb{R}\times\mathbb{S}(-\pi,\pi))\to\mathbb{K}\setminus(-\infty,0]$
is a homeomeorphism. ∎

As pointed out in the Introduction, even though $\mathbb{K}\setminus\\{0\\}$
is simply connected, in general, given a path in $\mathbb{K}\setminus\\{0\\}$,
the existence of a lift of this path to $\mathcal{E}^{+}_{\mathbb{K}}$ is not
guaranteed. Indeed, consider the following examples.

######  Example 4.6.

__

  1. (a)

Let $\sigma:[\pi/2,3\pi/2]\to\mathbb{K}\setminus\\{0\\}$ be the path (depicted
in Figure 1) defined by

$\sigma(t)=\cos(t)+I(t)\sin(t)$

where $I:[\pi/2,3\pi/2]\to\mathbb{S}$ is defined as

$\,\,\,\,\,I(t)=i\quad\textnormal{for}\quad\pi/2\leq t<\pi\quad\textnormal{and
}\quad I(t)=-j\quad\textnormal{for}\quad\pi\leq t\leq 3\pi/2.$

The curve $\sigma$ is continuous, but the function

$\textnormal{Arg}(\sigma(t))=I(t)\arg_{I}(\sigma(t))$

is not continuous at $\pi$ (the left and right limits are different).
Therefore $\sigma$ cannot be lifted to $\mathcal{E}^{+}_{\mathbb{K}}$.

Figure 1. The arc $\sigma\qquad\qquad\qquad$

  2. (b)

Consider now the loop $\gamma:[0,1]\to\mathbb{K}\setminus\\{0\\}$ defined by

$\gamma(t)=\cos(\pi-2\pi t)+t(1-t)(i\cos(2\pi/t)+j\sin(2\pi/t)),$

where $i,j$ are the usual orthogonal imaginary units (see Figure 2). Notice
that the imaginary part of $\gamma$ is continuous at all points of the
interval $[0,1]$ (including $0$). Nevertheless, for $t$ near to $0$, we have
that the function

$\displaystyle\textnormal{Arg}(\gamma(t))=$
$\displaystyle=(i\cos(2\pi/t)+j\sin(2\pi/t))\arccos\left(\frac{\cos(\pi-2\pi
t)}{\sqrt{\cos^{2}(\pi-2\pi t)+t^{2}(1-t)^{2}}}\right)$

has no limit for $t$ approaching $0^{+}$. Therefore $\gamma$ cannot be lifted
to $\mathcal{E}^{+}_{\mathbb{K}}$.

  3. (c)

Notice that in both the preceding cases, the paths
$\hat{\sigma}:=-\overline{\sigma}$ and $\hat{\gamma}:=-\overline{\gamma}$ can
be lifted to $\mathcal{E}^{+}_{\mathbb{K}}$, since their images are included
in $\mathbb{K}\setminus(-\infty,0]$ (see Proposition 4.5).

Figure 2. The path $\gamma$ (negative rocket) of the Example 4.6 (b) is drawn
on the left: it cannot be lifted to $\mathcal{E}^{+}_{\mathbb{K}}$. Its
reflection on the right (positive rocket) can be lifted to
$\mathcal{E}^{+}_{\mathbb{K}}$.

It is useful to point out that the existence of a lift $\Gamma$ of a path
$\gamma$ to $\mathcal{E}_{\mathbb{K}}^{+}$ is equivalent to the existence of a
continuous function $\textnormal{Arg}^{\gamma}:[a,b]\rightarrow{\rm
Im}(\mathbb{H})$, such that

$\Gamma(t)=(\gamma(t),\textnormal{Arg}^{\gamma}(t))\in\mathcal{E}^{+}_{\mathbb{K}}.$

As noticed in Remark 2.1, the function $\textnormal{Arg}^{\gamma}$ will be
decomposed, where possible, with obvious notation, as

$\textnormal{Arg}^{\gamma}:=\mathcal{I}^{\gamma}\arg^{\gamma}$

where $\mathcal{I}^{\gamma}:[a,b]\rightarrow\mathbb{S}$ and
$\arg^{\gamma}:[a,b]\rightarrow\mathbb{R}$. The existence of
$\mathcal{I}^{\gamma}:[a,b]\rightarrow\mathbb{S}$ implies that we can assign
to each $t\in[a,b]$ a complex plane $\mathbb{C}_{\mathcal{I}^{\gamma}}$ which
contains the point $\gamma(t)$ and hence determines the argument up to a
multiple of $2\pi.$

Complex slices $\\{\mathbb{C}_{J}\\}_{J\in\mathbb{S}}$ are naturally
parameterized by the elements of $\mathbb{S}/\\{\pm\operatorname{Id}\\}$, the
real projective space $\mathbb{R}\mathbb{P}^{\dim_{\mathbb{R}}\mathbb{K}-2}$
of dimension $\dim\mathbb{K}-2$. The projection $[\
]:\mathbb{S}\rightarrow\mathbb{S}/\\{\pm\operatorname{Id}\\}=\mathbb{R}\mathbb{P}^{\dim_{\mathbb{R}}\mathbb{K}-2}$
is the classical $2:1$ universal covering map and, as customary, for
$J\in\mathbb{S}$, the symbol $[J]$ denotes the equivalence class whose
representatives are the opposite (conjugate) imaginary units
$J,-J\in\mathbb{S}.$ Each element
$[J]\in\mathbb{S}/\\{\pm\operatorname{Id}\\}$ uniquely defines the complex
slice $\mathbb{C}_{[J]}=\mathbb{C}_{J}=\mathbb{C}_{-J}$. A continuous
imaginary unit function $\mathcal{I}^{\gamma}:[a,b]\to\mathbb{S}$ naturally
defines a continuous function
$\mathfrak{I}^{\gamma}:[a,b]\to\mathbb{S}/\\{\pm\operatorname{Id}\\}$ when we
set $\mathfrak{I}^{\gamma}(t)=[\mathcal{I}^{\gamma}(t)].$

######  Definition 4.7.

. Let $[a,b]\subset\mathbb{R}$ and let
$\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a path.

A path
$\mathfrak{I}^{\gamma}:[a,b]\rightarrow\mathbb{S}/\\{\pm\operatorname{Id}\\}$
such that $\gamma(t)\in\mathbb{C}_{\mathfrak{I}^{\gamma}(t)}$ for every
$t\in[a,b]$ is called a _companion_ of the path $\gamma$.

If a companion $\mathfrak{I}^{\gamma}$ of the path $\gamma$ exists, then
$\gamma$ is called a _path with a companion_ and the pair
$(\gamma,\mathfrak{I}^{\gamma})$ is called a _path with companion_
$\mathfrak{I}^{\gamma}$.

If the path $\gamma$ has a unique companion $\mathfrak{I}^{\gamma}$, then both
$\gamma$ and the pair $(\gamma,\mathfrak{I}^{\gamma})$ are called a _tame_
path.

######  Proposition 4.8.

Let $\gamma:[a,b]\to\mathbb{K}\setminus\\{0\\}$ be a path with companion
$\mathfrak{I}^{\gamma}$. If
$\mathcal{I}^{\gamma},-\mathcal{I}^{\gamma}:[a,b]\to\mathbb{S}$ are the two
lifts of of $\mathfrak{I}^{\gamma}$, then there exist continuous functions
$x,y:[a,b]\to\mathbb{R}$ such that, for all $t\in[a,b]$,

$\gamma(t)=x(t)+\mathcal{I}^{\gamma}(t)y(t)=x(t)+(-\mathcal{I}^{\gamma}(t))(-y(t)).$

These last expressions are called _canonical forms of_
$(\gamma,\mathfrak{I}^{\gamma})$.

###### Proof.

Since $\gamma(t)$ and $\mathcal{I}^{\gamma}(t)$ are both continuous, then
${\rm Re}(\gamma(t))=x(t)$ and $-\mathcal{I}^{\gamma}(t){\rm
Im}(\gamma(t))=y(t)$ are continuous as well on $[a,b]$. ∎

It is easy to see that all paths lying entirely in a complex slice have a
companion. Notice as well that a path
$\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ may have more than one
companion: this happens for example when the path $\gamma$ is such that
$\gamma([a,b])\subset(0,\infty)$; in this case, for an arbitrary path
$\mathcal{I}^{\gamma}:[a,b]\to\mathbb{S}$, the induced path
$\mathfrak{I}^{\gamma}:[a,b]\rightarrow\mathbb{S}/\\{\pm\operatorname{Id}\\}$
is a companion of $\gamma$; consequently $\gamma$ is not tame. For a similar
reason, a path $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ which maps
a closed sub-interval of $[a,b]$ to a real number has more than one companion,
and hence is not tame.

######  Remark 4.9.

_There exist paths in $\mathbb{K}\setminus\\{0\\}$ which can be lifted to
$\mathcal{E}^{+}_{\mathbb{K}}$, but have no companion. Indeed, set_

$\hat{\sigma}=-\overline{\sigma}:[\pi/2,3\pi/2]\to\mathbb{K}\setminus\\{0\\}$

_where $\sigma$ is the path defined in Example 4.6 (a). The path
$\hat{\sigma}$ is the symmetric image of the path $\sigma$ with respect to the
plane of purely imaginary quaternions (see figure 1) and, as pointed out in
Example 4.6 (c), it can be lifted to $\mathcal{E}^{+}_{\mathbb{K}}$. Obviously
$\hat{\sigma}$ has no companion: the continuity of a companion cannot hold at
$t=\pi$. _

The following definition will play a central role in the sequel.

######  Definition 4.10.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a path with a
companion
$\mathfrak{I}^{\gamma}:[a,b]\to\mathbb{S}/\\{\pm\operatorname{Id}\\}$, let
$\mathcal{I}^{\gamma},-{\mathcal{I}^{\gamma}}:[a,b]\to\mathbb{S}$ be the two h
(continuous) lifts of $\mathfrak{I}^{\gamma}$ to $\mathbb{S}$ and let

$\gamma(t)=x(t)+\mathcal{I}^{\gamma}(t)y(t)=x(t)+(-\mathcal{I}^{\gamma})(t)(-y(t))$

be the canonical forms of $(\gamma,\mathfrak{I}^{\gamma})$. The paths
$\gamma_{\mathcal{I}^{\gamma}},\gamma_{-\mathcal{I}^{\gamma}}:[a,b]\to\mathbb{C}\setminus\\{0\\}$
defined by

$\gamma_{\mathcal{I}^{\gamma}}=x(t)+iy(t),\qquad\qquad\gamma_{-\mathcal{I}^{\gamma}}=x(t)-iy(t)$

are called _the (two conjugated) shadows associated with the pair
$(\gamma,\mathfrak{I}^{\gamma})$_. If the path $\gamma$ is tame, then the
paths $\gamma_{\mathcal{I}^{\gamma}}$ and $\gamma_{-\mathcal{I}^{\gamma}}$ are
simply called _the (two) shadows associated with the path_ $\gamma$.

######  Remark 4.11.

_The two shadows associated with the pair ( $\gamma$, $\mathfrak{I}^{\gamma}$)
are conjugate paths._

Paths with a companion are of interest because they can all be lifted to
$\mathcal{E}^{+}_{\mathbb{K}}$.

######  Proposition 4.12.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a path with
companion
$\mathfrak{I}^{\gamma}:[a,b]\rightarrow\mathbb{S}/\\{\pm\operatorname{Id}\\}$.
Then there exist

  * •

a path $\mathcal{I}^{\gamma}:[a,b]\to\mathbb{S}$ with
$[\mathcal{I}^{\gamma}(t)]=\mathfrak{I}^{\gamma}(t)$, for all $t\in[a,b]$,

  * •

a path $\arg^{\gamma}:[a,b]\to\mathbb{R}$,

such that, after setting
$\textnormal{Arg}^{\gamma}=\mathcal{I}^{\gamma}\arg^{\gamma}:[a,b]\to{\rm
Im}(\mathbb{H})$, the path

$\Gamma(t)=(\gamma(t),\textnormal{Arg}^{\gamma}(t))$

is a lift of $\gamma$ to $\mathcal{E}^{+}_{\mathbb{K}}$ with
$\arg^{\gamma}(a)\in[0,\pi]$, called a _$\mathfrak{I}^{\gamma}$ -lift of
$\gamma$_.

If, as in Definition 3.5, for every $k\in\mathbb{Z}$ we set
$\arg^{\gamma}_{2k}:=\arg^{\gamma}+2k\pi$ and
$\textnormal{Arg}_{2k}^{\gamma}:=\mathcal{I}^{\gamma}\arg_{2k}^{\gamma}$, then
the path

$\Gamma_{k}(t)=(\gamma(t),\textnormal{Arg}_{2k}^{\gamma}(t))$

is a $\mathfrak{I}^{\gamma}$-lift of $\gamma$ to
$\mathcal{E}^{+}_{\mathbb{K}}$ with
$\arg^{\gamma}_{2k}(a)\in[2k\pi,(2k+1)\pi]$.

###### Proof.

There exist exactly two continuous lifts
$\mathcal{I}^{\gamma},-\mathcal{I}^{\gamma}$ of $\mathfrak{I}^{\gamma}$ to the
universal covering $\mathbb{S}$ of $\mathbb{S}/\\{\pm\operatorname{Id}\\}$.
Correspondingly, there exist two shadows
$\gamma_{\mathcal{I}^{\gamma}},\gamma_{-\mathcal{I}^{\gamma}}:[a,b]\to\mathbb{C}\setminus\\{0\\}$
associated with $\mathfrak{I}^{\gamma}$. Exchange $\mathcal{I}^{\gamma}$ and
$-\mathcal{I}^{\gamma}$ if necessary, so that $\mathcal{I}^{\gamma}$ is such
that $\arg(\gamma_{\mathcal{I}^{\gamma}}(a))\in[0,\pi]$. As a complex path,
$\gamma_{\mathcal{I}^{\gamma}}$ has a well defined argument
$\arg^{\gamma_{\mathcal{I}^{\gamma}}}:[a,b]\to\mathbb{R}$ such that
$\arg^{\gamma_{\mathcal{I}^{\gamma}}}(a)\in[0,\pi]$. Set
$\arg^{\gamma}:=\arg^{\gamma_{\mathcal{I}^{\gamma}}}$. Then the chosen paths
$\mathcal{I}^{\gamma}$ and $\arg^{\gamma}$ have the properties required in the
statement. The rest of the proof is straightforward. ∎

The lifts $\Gamma$ and $\Gamma_{k}$ (for $k\in\mathbb{Z}$) appearing in the
last Proposition are not unique, when $\Gamma(a)$ and $\Gamma_{k}(a)$ are
real.

At this point, Proposition 4.2 implies directly the existence of all branches
of the logarithm, along all paths in $\mathbb{K}\setminus\\{0\\}$ having a
companion.

######  Corollary 4.13.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a path with
companion
$\mathfrak{I}^{\gamma}:[a,b]\rightarrow\mathbb{S}/\\{\pm\operatorname{Id}\\}$.
For every $k\in\mathbb{Z}$, let

$\Gamma_{k}=(\gamma,\textnormal{Arg}_{2k}^{\gamma})$

be a $\mathfrak{I}^{\gamma}$-lift of $\gamma$ to
$\mathcal{E}^{+}_{\mathbb{K}}$ with
$\arg^{\gamma}_{2k}(a)\in[2k\pi,(2k+1)\pi]$. Then, the map defined on the
interval $[a,b]$ by

$(\log_{k}\circ\gamma)(t):=(L\circ\Gamma_{k})(t)=\log|\gamma(t)|+\textnormal{Arg}_{2k}(\gamma(t))$

is a continuation of the hypercomplex logarithm along $\gamma$ with initial
point $\log|\gamma(a)|+\textnormal{Arg}_{2k}(\gamma(a))$. This map is called
_a $k$-th branch of the hypercomplex logarithm along $\gamma$ with initial
point $\log|\gamma(a)|+\textnormal{Arg}_{2k}(\gamma(a))$._

###### Proof.

The proof is a straightforward consequence of Proposition 4.12 and Proposition
4.2. ∎

We will now turn our attention to the case of loops of
$\mathbb{K}\setminus\\{0\\}$.

######  Definition 4.14.

Let $[a,b]\subset\mathbb{R}$ and let
$\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a path with a companion
$\mathfrak{I}^{\gamma}:[a,b]\rightarrow\mathbb{S}/\\{\pm\operatorname{Id}\\}$.

If both $\gamma$ and $\mathfrak{I}^{\gamma}$ are closed, then the path
$\gamma$ is called _a loop with companion_ $\mathfrak{I}^{\gamma}$, and the
pair $(\gamma,\mathfrak{I}^{\gamma})$ is called _a loop with companion_.

The loop with companion $(\gamma,\mathfrak{I}^{\gamma})$ is called _untwisted_
if $\mathfrak{I}^{\gamma}$ is homotopic to a constant in
$\mathbb{S}/\\{\pm\operatorname{Id}\\}$; if instead $\mathfrak{I}^{\gamma}$ is
not homotopic to a constant, then $(\gamma,\mathfrak{I}^{\gamma})$ is said to
be _twisted_.

In the most relevant case in which $\gamma$ is tame, we can specialize the
definition as follows.

######  Definition 4.15.

Let $[a,b]\subset\mathbb{R}$ and let
$\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a tame path with
companion
$\mathfrak{I}^{\gamma}:[a,b]\rightarrow\mathbb{S}/\\{\pm\operatorname{Id}\\}$.

If both $\gamma$ and $\mathfrak{I}^{\gamma}$ are closed, then $\gamma$ is
called _a tame loop (with companion_ $\mathfrak{I}^{\gamma}$), and the pair
$(\gamma,\mathfrak{I}^{\gamma})$ is called _a tame loop_.

The tame loop $(\gamma,\mathfrak{I}^{\gamma})$ is called _untwisted_ if
$\mathfrak{I}^{\gamma}$ is homotopic to a constant in
$\mathbb{S}/\\{\pm\operatorname{Id}\\}$; if instead $\mathfrak{I}^{\gamma}$ is
not homotopic to a constant, then $(\gamma,\mathfrak{I}^{\gamma})$ is said to
be _twisted_.

######  Remark 4.16.

For any fixed $I\in\mathbb{S}$, let
$\gamma:[a,b]\to\mathbb{K}\setminus\\{0\\}$ be a path lying in the complex
slice $\mathbb{C}_{I}$. The path $\gamma$ has always a particularly simple
companion, namely
$\mathfrak{I}^{\gamma}:[a,b]\to\mathbb{S}/\\{\pm\operatorname{Id}\\}$
constantly equal to $[I]$. Moreover, the two different lifts of
$\mathfrak{I}^{\gamma}$ to $\mathbb{S}$ are both constantly equal to $I$ or
$-I$, respectively. As a consequence, if the given path $\gamma$ is closed and
tame, it is a tame loop and is untwisted.

A twisted loop necessarily intersects the real axis. Indeed the following
result holds.

######  Proposition 4.17.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a loop which misses
the real axis. Then $\gamma$ is a tame loop and is untwisted.

###### Proof.

By Proposition 4.5, the loop $\gamma$ can be lifted to a path
$\Gamma:[a,b]\to\mathcal{E}^{+}_{\mathbb{K}}$ with
$\Gamma=(\gamma,\textnormal{Arg}^{\gamma})$. Let us consider the map
$\textnormal{Arg}^{\gamma}=\mathcal{I}^{\gamma}\arg^{\gamma}:[a,b]\to{\rm
Im}(\mathbb{K})$. By the hypothesis, there exists $k\in\mathbb{Z}$ such that
the map $\arg^{\gamma}=\arg^{\gamma}_{2k}:[a,b]\to(2k\pi,(2k+1)\pi)$ is never
vanishing and hence has constant sign. Now, since $\gamma$ is closed, we have
that

$\mathcal{I}^{\gamma}(a)\arg^{\gamma}_{2k}(a)=\mathcal{I}^{\gamma}(b)\arg^{\gamma}_{2k}(b).$

Since $\arg^{\gamma}_{2k}(a)$ and $\arg^{\gamma}_{2k}(b)$ have the same sign
and both belong to the interval $(2k\pi,(2k+1)\pi)$, we obtain

$\arg^{\gamma}_{2k}(a)=\arg^{\gamma}_{2k}(b)$

and hence

$\mathcal{I}^{\gamma}(a)=\mathcal{I}^{\gamma}(b).$

Therefore the path $\mathcal{I}^{\gamma}:[a,b]\to\mathbb{S}$ is a loop, and
hence the unique companion
$[\mathcal{I}^{\gamma}]:[a,b]\to\mathbb{S}/\\{\pm\operatorname{Id}\\}$ is a
loop, homotopic to a constant. As a consequence the path $\gamma$ is an
untwisted, tame loop. ∎

###  4.1. Winding number for untwisted loops with companion in
$\mathbb{K}\setminus\\{0\\}$

It is well known that the definition of winding number for a loop (with
respect to a point) is not natural in $\mathbb{R}^{n}$ when $n$ is greater
than $2$. Nevertheless, in our setting, we can start by giving a definition of
winding number that has full meaning for loops with companion that are
untwisted and lie in $\mathbb{K}\setminus\\{0\\}$.

The following result opens a way to this definition of winding number.

######  Proposition 4.18.

A loop $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\},$
$\gamma([a,b])\not\subset\mathbb{R}$, with companion $\mathfrak{I}^{\gamma}$
is untwisted if, and only if, for any chosen non real initial point of
$\gamma$, both shadows associated with $\mathfrak{I}^{\gamma}$ are loops.

###### Proof.

Let $\gamma_{\mathcal{I}^{\gamma}}:[a,b]\to\mathbb{C}\setminus\\{0\\}$,
$\gamma_{\mathcal{I}^{\gamma}}(t)=x(t)+iy(t),$ be one of the shadows
associated with $\mathfrak{I}^{\gamma}$.

If the loop $\gamma$ is untwisted, then any lift $\mathcal{I}^{\gamma}$ of the
companion $\mathfrak{I}^{\gamma}$ of $\gamma$ is a loop, and hence it has
coinciding endpoints. Therefore, the path

$\gamma(t)=x(t)+\mathcal{I}^{\gamma}(t)y(t)$

being a loop, the continuous function $y:[a,b]\to\mathbb{R}$ is such that
$y(a)=y(b)$. Hence the associated shadow
$\gamma_{\mathcal{I}^{\gamma}}(t)=x(t)+iy(t)$ is closed.

On the other hand, suppose the associated shadow
$\gamma_{\mathcal{I}^{\gamma}}:[a,b]\to\mathbb{C}\setminus\\{0\\}$,
$\gamma_{\mathcal{I}^{\gamma}}(t)=x(t)+iy(t)$, is a loop and assume that
$y(a)=y(b)\neq 0.$ Since the path

$\gamma(t)=x(t)+\mathcal{I}^{\gamma}(t)y(t)$

is a loop by assumption, we obtain
$\mathcal{I}^{\gamma}(a)=\mathcal{I}^{\gamma}(b)$ and so the lift
$\mathcal{I}^{\gamma}$ of $\mathfrak{I}^{\gamma}$ is a loop. In conclusion,
$\gamma$ is untwisted. ∎

We are now ready to use the well established definition of winding number for
complex loops in $\mathbb{C}\setminus\\{0\\}$ to define the winding number in
the case of untwisted loops with companion in $\mathbb{K}\setminus\\{0\\}$.

######  Definition 4.19.

Let the loop $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ with
companion $\mathfrak{I}^{\gamma}$ be untwisted. The winding number (with
respect to zero) of the loop $(\gamma,\mathfrak{I}^{\gamma})$, denoted
$\emph{wind}(\gamma,\mathfrak{I}^{\gamma})$, is defined as the absolute value
of the winding number (with respect to zero),
$\emph{wind}(\gamma_{\mathcal{I}^{\gamma}})$, of a shadow
$\gamma_{\mathcal{I}^{\gamma}}$ associated with $\mathfrak{I}^{\gamma}$:

$\emph{wind}(\gamma,\mathfrak{I}^{\gamma})=|\emph{wind}(\gamma_{\mathcal{I}^{\gamma}})|.$

In the case in which the loop $(\gamma,\mathfrak{I}^{\gamma})$ is tame, there
is one and only one companion of $\gamma$, and hence we can simply denote the
winding number of $\gamma$ by $\emph{wind}(\gamma)$.

Of course, we need to show that the given definition of winding number of an
untwisted loop with companion $(\gamma,\mathfrak{I}^{\gamma})$ does not depend
on the choice of the shadow associated with $\mathfrak{I}^{\gamma}$. Indeed,
the two shadows associated with $\mathfrak{I}^{\gamma}$ are conjugate loops:
as a consequence, their winding numbers are opposite. Therefore, Definition
4.19 is consistent.

One of the important features of the classical winding number (with respect to
zero) of loops of $\mathbb{C}\setminus\\{0\\}$ is its invariance with respect
to homotopy between such loops. The winding number of an untwisted loop with
companion (in $\mathbb{K}\setminus\\{0\\}$) just defined cannot be invariant
with respect to standard homotopy in $\mathbb{K}\setminus\\{0\\}$: all such
loops are homotopic to a constant loop since $\mathbb{K}\setminus\\{0\\}$ is
simply connected, and a constant loop has vanishing winding number.

A special notion of homotopy comes into the scenery in our setting. The next
definition is useful to define such a notion.

######  Definition 4.20.

Let $[a,b]\times[c,d]\subset\mathbb{R}^{2}$ and let
$F:[a,b]\times[c,d]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a continuous map.

A continuous map
$\mathfrak{I}^{F}:[a,b]\times[c,d]\rightarrow\mathbb{S}/\\{\pm\operatorname{Id}\\}$
such that $F(t,s)\in\mathbb{C}_{\mathfrak{I}^{F}(t,s)}$ for every
$(t,s)\in[a,b]\times[c,d]$ is called a _companion_ of the map $F$.

If a companion $\mathfrak{I}^{F}$ of the map $F$ exists, then $F$ is called a
_continuous map with companion_ $\mathfrak{I}^{F}$, and $(F,\mathfrak{I}^{F})$
is called a _continuous map with companion._

If the map $F$ has a unique companion $\mathfrak{I}^{F}$, then it is called a
_tame_ map.

######  Proposition 4.21.

Let $F:[a,b]\times[c,d]\to\mathbb{K}\setminus\\{0\\}$ be a continuous map with
companion $\mathfrak{I}^{F}$. If
$\mathcal{I}^{F},-\mathcal{I}^{F}:[a,b]\times[c,d]\to\mathbb{S}$ are the two
lifts of $\mathfrak{I}^{F}$, then there exist continuous functions
$x,y:[a,b]\times[c,d]\to\mathbb{R}$ such that, for all
$(t,s)\in[a,b]\times[c,d]$,

$F(t)=x(t,s)+\mathcal{I}^{F}(t,s)y(t,s)=x(t,s)+(-\mathcal{I}^{F}(t,s))(-y(t,s)).$

These last expressions are called _canonical forms of_ $(F,\mathfrak{I}^{F})$.

###### Proof.

See the proof of Proposition 4.8 ∎

As announced, the idea is now to define a special type of homotopy between
paths, each having a companion and sharing the same endpoints. As customary,
also in this paper homotopy between paths will always be meant with fixed
endpoints.

######  Definition 4.22.

Let $\gamma_{1},\gamma_{2}:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be two
paths with the same endpoints $\gamma_{1}(a)=\gamma_{2}(a)=p$ and
$\gamma_{1}(b)=\gamma_{2}(b)=q$. Let $\mathfrak{I}^{\gamma_{1}}$ and
$\mathfrak{I}^{\gamma_{2}}$ be companions of $\gamma_{1}$ and $\gamma_{2}$
respectively. If there exists a continuous map
$F:[a,b]\times[0,1]\rightarrow\mathbb{K}\setminus\\{0\\}$ with companion
$\mathfrak{I}^{F}$ such that:

  1. (1)

$\mathfrak{I}^{F}(t,0)=\mathfrak{I}^{\gamma_{1}}(t)$ and
$\mathfrak{I}^{F}(t,1)=\mathfrak{I}^{\gamma_{2}}(t)$, for all $t\in[a,b]$;

  2. (2)

$F(t,0)=\gamma_{1}(t)$ and $F(t,1)=\gamma_{2}(t)$, for all $t\in[a,b]$;

  3. (3)

$F(0,s)=p$ and $F(1,s)=q$, for all $s\in[0,1]$;

then we will say that
$(\gamma_{1},\mathfrak{I}^{\gamma_{1}}),(\gamma_{2},\mathfrak{I}^{\gamma_{2}})$
are _companion homotopic (or c-homotopic)_ and that $(F,\mathfrak{I}^{F})$ is
a _c-homotopy_ between $(\gamma_{1},\mathfrak{I}^{\gamma_{1}})$ and
$(\gamma_{2},\mathfrak{I}^{\gamma_{2}})$.

Let $\gamma_{1},\gamma_{2}:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be two
paths with the same endpoints. If there exist a companion
$\mathfrak{I}^{\gamma_{1}}$ of $\gamma_{1}$ and a companion
$\mathfrak{I}^{\gamma_{2}}$ of $\gamma_{2}$ such that
$(\gamma_{1},\mathfrak{I}^{\gamma_{1}}),(\gamma_{2},\mathfrak{I}^{\gamma_{2}})$
are c-homotopic, then we say that $\gamma_{1}$ and $\gamma_{2}$ are _weakly
c-homotopic_.

The following simple result will be helpful in the sequel.

######  Proposition 4.23.

Let the continuous map $F:[a,b]\times[c,d]\to\mathbb{K}\setminus\\{0\\}$ with
companion $\mathfrak{I}^{F}$ be a c-homotopy between
$(\gamma_{1},\mathfrak{I}^{\gamma_{1}})$ and
$(\gamma_{2},\mathfrak{I}^{\gamma_{2}})$. Then:

  1. (i)

the map $\mathfrak{I}^{F}$ is a homotopy between $\mathfrak{I}^{\gamma_{1}}$
and $\mathfrak{I}^{\gamma_{2}}$;

  2. (ii)

the homotopy $\mathfrak{I}^{F}$ can be lifted to a homotopy $\mathcal{I}^{F}$
between a lift $\mathcal{I}^{\gamma_{1}}$ of $\mathfrak{I}^{\gamma_{1}}$ and a
lift $\mathcal{I}^{\gamma_{2}}$ of $\mathfrak{I}^{\gamma_{2}}$ in such a way
that the canonical form of $F$

(4.3) $F(t,s)=x(t,s)+\mathcal{I}^{F}(t,s)y(t,s)$

is a homotopy between the canonical forms

$\gamma_{1}(t)=x_{1}(t)+\mathcal{I}^{\gamma_{1}}(t)y_{1}(t)$

and

$\gamma_{2}(t)=x_{2}(t)+\mathcal{I}^{\gamma_{2}}(t)y_{2}(t)$

of $\gamma_{1}$ and $\gamma_{2}$, respectively;

  3. (iii)

the shadows

$x_{1}(t)+iy_{1}(t),\qquad x_{2}+iy_{2}(t)$

of $(\gamma_{1},\mathfrak{I}^{\gamma_{1}})$ and
$(\gamma_{2},\mathfrak{I}^{\gamma_{2}})$, respectively, are homotopic in
$\mathbb{C}\setminus\\{0\\}$.

###### Proof.

The proofs of _(i)_ and _(ii)_ are a straightforward consequence of Definition
4.22 and of what is stated in Propositions 4.8 and 4.21. Let us prove _(iii)_.
To this aim, consider the canonical form of $F$ that appears in (4.3) and the
following continuous maps, for $(t,s)\in[a,b]\times[0,1]$:

$\begin{array}[]{l}\mathcal{L}_{1}(t,s)={\rm Re}(F(t,s)),\\\
\mathcal{L}_{2}(t,s)=-\mathcal{I}^{F}(t,s){\rm Im}(F(t,s))\end{array}$

We will prove that
$\mathcal{L}=(\mathcal{L}_{1},\mathcal{L}_{2}):[a,b]\times[0,1]\to\mathbb{R}^{2}\setminus\\{(0,0)\\}$
is a homotopy between the two given shadows of $\gamma_{1}$ and $\gamma_{2}$.
Indeed, using directly formula (4.3) for the canonical form of $F$, it is easy
to check that on $[a,b]\times[0,1]$,

$\begin{array}[]{l}\mathcal{L}(t,0)=(\mathcal{L}_{1}(t,0),\mathcal{L}_{2}(t,0))=(x_{1}(t),y_{1}(t)),\\\
\mathcal{L}(t,1)=(\mathcal{L}_{1}(t,1),\mathcal{L}_{2}(t,1))=(x_{2}(t),y_{2}(t)),\\\
\mathcal{L}(a,s)=(\mathcal{L}_{1}(a,s),\mathcal{L}_{2}(a,s))=(x_{1}(a),y_{1}(a))=(x_{2}(a),y_{2}(a)),\\\
\mathcal{L}(b,s)=(\mathcal{L}_{1}(b,s),\mathcal{L}_{2}(b,s))=(x_{1}(b),y_{1}(b)=(x_{2}(b),y_{2}(b)).\\\
\end{array}$

The proof is now complete.

∎

######  Example 4.24.

To better illustrate the major difference between complex and quaternionic
cases, consider the complex curve, defined by

$\begin{array}[]{ll}\gamma(t)=3e^{it},\,t\in[0,\pi],&\gamma(t)=-4+\dfrac{t}{\pi},\,t\in[\pi,3\pi],\\\
\\\
\gamma(t)=e^{-it},\,t\in[3\pi,4\pi],&\gamma(t)=-3+\dfrac{t}{\pi},\,t\in[4\pi,6\pi].\end{array}$

As a complex curve, i.e. with the constant companion $i,$ the curve $\gamma$
has winding number $0$ and coincides with its own shadow. As a quaternionic
curve, $\gamma$ has a large family of companions $\mathfrak{I}^{\gamma}$; for
example one can consider-

$\begin{array}[]{ll}\mathcal{I}^{\gamma}(t)=i,\,t\in[0,\pi],&\mathcal{I}^{\gamma}(t)=\mathcal{J}(t),\,t\in[\pi,3\pi],\\\
\mathcal{I}^{\gamma}(t)=-i,\,t\in[3\pi,4\pi],&\mathcal{I}^{\gamma}(t)=-\mathcal{J}(t),\,t\in[4\pi,6\pi];\end{array}$

and $\mathfrak{I}^{\gamma}(t)=[\mathcal{I}^{\gamma}(t)]$, where
$\mathcal{J}:[\pi,3\pi]\rightarrow\mathbb{S}$ is an arbitrary continuous curve
with $\mathcal{J}(\pi)=i$ and $\mathcal{J}(3\pi)=-i.$ Correspondingly, the
shadow of $(\gamma,\mathfrak{I}^{\gamma})$ is

$\begin{array}[]{ll}\gamma_{\mathfrak{I}^{\gamma}}(t)=3e^{it},\,t\in[0,\pi],&\gamma_{\mathfrak{I}^{\gamma}}(t)=-4+\dfrac{t}{\pi},\,t\in[\pi,3\pi],\\\
\\\
\gamma_{\mathfrak{I}^{\gamma}}(t)=e^{it},\,t\in[3\pi,4\pi],&\gamma_{\mathfrak{I}^{\gamma}}(t)=-3+\dfrac{t}{\pi},\,t\in[4\pi,6\pi]\end{array}$

and so the winding number of $(\gamma,\mathfrak{I}^{\gamma})$ is $1.$ The
pairs $(\gamma,i)$ and $(\gamma,\mathfrak{I}^{\gamma})$ are not c-homotopic.

The notion of c-homotopy is particularly useful in this setting, because of
the following result.

######  Proposition 4.25.

Let
$(\gamma_{1},\mathfrak{I}^{\gamma_{1}}),(\gamma_{2},\mathfrak{I}^{\gamma_{2}}):[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$
be two paths with companions and with the same endpoints
$\gamma_{1}(a)=\gamma_{2}(a)=p$ and $\gamma_{1}(b)=\gamma_{2}(b)=q$. Then the
following statements are equivalent :

  1. (1)

$(\gamma_{1},\mathfrak{I}^{\gamma_{1}}),(\gamma_{2},\mathfrak{I}^{\gamma_{2}})$
are c-homotopic;

  2. (2)

$\mathfrak{I}^{\gamma_{1}}$ and $\mathfrak{I}^{\gamma_{2}}$ are homotopic in
$\mathbb{S}/\left\\{\pm Id\right\\}$, and, in addition, for each of the
shadows of $(\gamma_{1},\mathfrak{I}^{\gamma_{1}})$ there is a shadow of
$(\gamma_{2},\mathfrak{I}^{\gamma_{2}})$ so that these two shadows are
homotopic in $\mathbb{C}\setminus\\{0\\}$.

###### Proof.

Suppose first that _(2)_ holds. Then there exist:

  * •

a homotopy $\mathbb{G}:[a,b]\times[0,1]\to\mathbb{S}/\left\\{\pm Id\right\\}$
between $\mathfrak{I}^{\gamma_{1}}$ and $\mathfrak{I}^{\gamma_{2}}$;

  * •

a lift of $\mathbb{G}$, i.e. a homotopy $G:[a,b]\times[0,1]\to\mathbb{S}$
between a lift $\mathcal{I}^{\gamma_{1}}$ of $\mathfrak{I}^{\gamma_{1}}$ and a
lift $\mathcal{I}^{\gamma_{2}}$ of $\mathfrak{I}^{\gamma_{2}}$;

  * •

a homotopy
$L=(L_{1},L_{2}):[a,b]\times[0,1]\to\mathbb{R}^{2}\setminus\\{(0,0)\\}$
between a shadow of $\gamma_{1}$ and a shadow of $\gamma_{2}$ (its “conjugate”
being a homotopy between the corresponding conjugate shadows).

In this situation, the map $F:[a,b]\times[0,1]\to\mathbb{K}\setminus\\{0\\}$
defined by

$F(t,s)=L_{1}(t,s)+G(t,s)L_{2}(t,s)$

is a homotopy between $\gamma_{1}$ and $\gamma_{2}$. Indeed, $F$ is obviously
continuous, and such that, for all $t\in[a,b]$ and all $s\in[0,1]$,

$\begin{array}[]{l}F(t,0)=L_{1}(t,0)+G(t,0)L_{2}(t,0)=x_{1}(t)+\mathcal{I}^{\gamma_{1}}(t)y_{1}(t)=\gamma_{1}(t);\\\
F(t,1)=L_{1}(t,1)+G(t,1)L_{2}(t,1)=x_{2}(t)+\mathcal{I}^{\gamma_{2}}(t)y_{2}(t)=\gamma_{2}(t);\\\
F(a,s)=L_{1}(a,s)+G(a,s)L_{2}(a,s)=x_{1}(a)+\mathcal{I}^{\gamma_{1}}(a)y_{1}(a)=\gamma_{1}(a)=\gamma_{2}(a);\\\
F(b,s)=L_{1}(b,s)+G(b,s)L_{2}(b,s)=x_{1}(b)+\mathcal{I}^{\gamma_{1}}(b)y_{1}(b)=\gamma_{1}(b)=\gamma_{2}(b).\end{array}$

Moreover, the continuous map $G:[a,b]\times[0,1]\to\mathbb{S}$ defines, by
construction, a companion of $F$ given by

$\mathfrak{I}^{F}(t,s)=[G(t,s)]=\mathbb{G}(t,s)$

for all $(t,s)\in[a,b]\times[0,1]$. As a consequence, $(F,\mathfrak{I}^{F})$
is a c-homotopy between
$(\gamma_{1},\mathfrak{I}^{\gamma_{1}}),(\gamma_{2},\mathfrak{I}^{\gamma_{2}})$.

Let us now suppose that _(1)_ holds, i.e. that
$(\gamma_{1},\mathfrak{I}^{\gamma_{1}})$ and
$(\gamma_{2},\mathfrak{I}^{\gamma_{2}})$ are c-homotopic. In this case
$\mathfrak{I}^{\gamma_{1}}$ and $\mathfrak{I}^{\gamma_{2}}$ are homotopic by
definition, and the rest of the assertion follows from Proposition 4.23.

∎

######  Proposition 4.26.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a loop with
companion $\mathfrak{I}^{\gamma}$. Then $(\gamma,\mathfrak{I}^{\gamma})$ is
untwisted if, and only if, it is c-homotopic to one of its (closed and
conjugate) shadows in $\mathbb{C}_{\mathfrak{I}^{\gamma}(a)}$.

###### Proof.

If $(\gamma,\mathfrak{I}^{\gamma})$ is untwisted, then any lift
$\mathcal{I}^{\gamma}$ of the companion $\mathfrak{I}^{\gamma}$ with initial
point $\mathcal{I}^{\gamma}(a)$ is homotopic in $\mathbb{S}$ to the constant
loop $\mathcal{I}^{\gamma}(a)$, and therefore the loop $\gamma$ is c-homotopic
to its (closed) shadow in $\mathbb{C}_{\mathcal{I}^{\gamma}(a)}$ (see
Proposition 4.18). On the other hand, if the loop with companion
$(\gamma,\mathfrak{I}^{\gamma})$ is c-homotopic to its shadow, then the lift
of its companion $\mathfrak{I}^{\gamma}$ with initial point
$\mathcal{I}^{\gamma}(a)$ has to be homotopic in $\mathbb{S}$ to the constant
loop $\mathcal{I}^{\gamma}(a)$. As a consequence the loop
$(\gamma,\mathfrak{I}^{\gamma})$ is untwisted by definition. ∎

The notion of c-homotopy is suitable to comply with the meaning of the winding
number of loops in the setting of $\mathbb{K}\setminus\\{0\\}$. In this
panorama, all untwisted tame loops play a special role: any such a loop has an
“intrinsically defined” winding number that depends only on its geometric
properties. Indeed, we can state the following result.

######  Theorem 4.27.

Let $\gamma,\delta:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be two
untwisted, tame loops. Then $\gamma$ and $\delta$ are c-homotopic if, and only
if, $\emph{wind}(\gamma)=\emph{wind}(\delta)$.

###### Proof.

By Proposition 4.25, $\gamma$ and $\delta$ are c-homotopic if, and only if,
the unique companions $\mathfrak{I}^{\gamma}$ and $\mathfrak{I}^{\delta}$ are
homotopic and a shadow of $(\gamma,\mathfrak{I}^{\gamma})$ is homotopic to a
shadow of $(\delta,\mathfrak{I}^{\delta})$, in $\mathbb{C}\setminus\\{0\\}$.
According to Definition 4.19, the winding number of
$(\gamma,\mathfrak{I}^{\gamma})$ (or $(\delta,\mathfrak{I}^{\delta})$) is
defined as the absolute value of the winding number of one of the two (closed)
shadows of $(\gamma,\mathfrak{I}^{\gamma})$ (or
$(\delta,\mathfrak{I}^{\delta})$). Therefore the proof is a straightforward
consequence of the properties of the fundamental group
$\Pi_{1}(\mathbb{C}\setminus\\{0\\})\equiv\mathbb{Z}$, where the class of each
loop is determined by its winding number (with respect to zero). ∎

The given definition of winding number, which has particularly transparent
geometrical meanings, cannot be adopted as it is in the twisted case, due to
the two following results.

######  Proposition 4.28.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a loop with a
companion $\mathfrak{I}^{\gamma}$. Then $(\gamma,\mathfrak{I}^{\gamma})$ is
twisted if, and only if, for any chosen non real initial point of $\gamma$,
any shadow associated with $(\gamma,\mathfrak{I}^{\gamma})$ has conjugate
endpoints.

###### Proof.

Let $\gamma_{\mathcal{I}^{\gamma}}:[a,b]\to\mathbb{C}\setminus\\{0\\}$,
$\gamma_{\mathcal{I}^{\gamma}}(t)=x(t)+iy(t),$ be one of the shadows
associated with $\mathfrak{I}^{\gamma}$.

If the loop $(\gamma,\mathfrak{I}^{\gamma})$ is twisted, then the lift
$\mathcal{I}^{\gamma}$ is not closed, and hence it has opposite endpoints.
Therefore, the path

$\gamma(t)=x(t)+\mathcal{I}^{\gamma}(t)y(t)$

being closed, the continuous function $y:[a,b]\to\mathbb{R}$ is such that
$y(a)=-y(b)$. Hence the associated shadow
$\gamma_{\mathcal{I}^{\gamma}}(t)=x(t)+iy(t)$ has conjugate endpoints.

On the other hand, suppose the associated shadow
$\gamma_{\mathcal{I}^{\gamma}}:[a,b]\to\mathbb{C}\setminus\\{0\\}$,
$\gamma_{\mathcal{I}^{\gamma}}(t)=x(t)+iy(t)$, has conjugate nonreal
endpoints. Then $y(a)=-y(b)\neq 0$ and, the path

$\gamma(t)=x(t)+\mathcal{I}^{\gamma}(t)y(t)$

being closed by assumption, we obtain
$\mathcal{I}^{\gamma}(a)=-\mathcal{I}^{\gamma}(b)$ and so the lift
$\mathcal{I}^{\gamma}$ of $\mathfrak{I}^{\gamma}$ is not closed. In
conclusion, $(\gamma,\mathfrak{I}^{\gamma})$ is twisted. ∎

######  Corollary 4.29.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a loop with
companion $\mathfrak{I}^{\gamma}$. Then the two shadows associated with
$\mathfrak{I}^{\gamma}$ are closed if, and only if, the endpoints of $\gamma$
are real.

###### Proof.

The proof follows immediately from Proposition 4.28. ∎

We might be encouraged to think that, in the case of a loop with companion
which is twisted, we should first parameterise the loop in such a way that it
has real endpoints (see Proposition 4.17), and then use Definition 4.19.
Indeed, this approach gives a weird result, if tested, for instance, in the
case of the twisted, tame loop $\lambda$ presented in the next example.

######  Example 4.30.

_Consider the loop $\lambda$ in the hyperplane of $\mathbb{H}$ generated by
the orthogonal units $\\{1,i,j\\}$. The path consists of several arcs: the arc
of parabola $t+1+t^{2}(i+j),t\in[-1,1],$ the segments from $(2,1,0)$ to
$(2,1,1)$, from $(2,1,0)$ to $(0,1,0)$ and from $(0,0,1)$ to $(0,1,1),$ the
halfcircle $\cos t+i\sin t,t\in[\pi/2,3\pi/2]$ and the quarter of circle
$i\cos t+j\sin t,t\in[\pi/2,\pi].$ Let the orientation be such that it
coincides with the positive orientation of the halfcircle part in the plane
containing $1,i$. The path intersects the real axis at points $z=1$ and
$z=-1.$ _

$\begin{array}[]{ccc}\includegraphics[width=130.08731pt]{T1.eps}&\includegraphics[width=130.08731pt]{T3.eps}&\includegraphics[width=130.08731pt]{T2.eps}\\\
\mbox{\footnotesize(a)}&\mbox{\footnotesize(b)}&\mbox{\footnotesize(c)}\\\
&&\end{array}$

Figure 3. From left to right: (a) the path $\gamma$ and two of its shadows
(b), (c) aaaaaaaaaaaaaaaaa

In the previous example, the proposed winding number of the twisted, tame loop
$\lambda$ would be $1$ if the loop is parameterised with real endpoints equal
to $1\in\mathbb{R}$ (see Figure 3 (b)). On the other hand, the same loop
$\lambda$ parameterised with endpoints equal to $-1\in\mathbb{R}$ would have
winding number $0$ (see Figure 3 (c)). What we just illustrated clarifies that
a notion of winding number for twisted, tame loops in
$\mathbb{K}\setminus\\{0\\}$ (if it exists) has to be given by following a
different approach.

In the spirit of the above example and Proposition 4.28 the definition of the
winding number for a closed tame twisted loop
$\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ cannot be given by
considering the change of the argument since this depends on the choice of the
initial point.

Assume that $\gamma$ is a twisted loop in $\mathbb{K}\setminus\\{0\\}$ which
intersects both the positive and the negative real axis; let
$\gamma_{\mathcal{I}^{\gamma}}:[a,b]\to\mathbb{C}\setminus\\{0\\}$,
$\gamma_{\mathcal{I}^{\gamma}}(t)=x(t)+iy(t),$ be one of the shadows
associated with $\gamma$. Let $\arg^{\gamma}(t),t\in[a,b],$ be the
corresponding argument and choose the initial argument so that
$\arg^{\gamma}(a)\in[0,\pi].$ The set $\Delta=\\{\arg^{\gamma}(a)\mbox{ for
all possible initial points}\\}$ is an interval contained in $[0,\pi].$
Because the loop $\gamma$ is twisted, the argument at $b$ is
$\arg^{\gamma}(b)=2n\pi-\arg^{\gamma}(a)$ and hence

$\arg^{\gamma}(b)-\arg^{\gamma}(a)=2n\pi-2\arg^{\gamma}(a)$

and is not an integer multiple of $2\pi$ unless $\arg^{\gamma}(a)=0,\pi.$ Even
if we set the initial point to be real, so that the change of argument is
$2n\pi,$ the number $n$ can have more than one value as shown in the following
example.

######  Example 4.31.

Let $\gamma_{1}$ be the positively oriented unit circle with initial point
$-1$ and companion $i$ and define
$\gamma_{2}(t):=\cos(t)+i\sin(t)+j(\cos(t)+1),t\in[-\pi,\pi]$. Choose the lift
of the companion $\mathcal{I}_{2}$ of $\gamma_{2}$ so that
$\mathcal{I}_{2}(-\pi)=i$ and $\mathcal{I}_{2}(-\pi)=-i.$ Let
$\gamma=\gamma_{1}^{m}\cdot\gamma_{2}$ denote the loop composed first of $m$
copies of $\gamma_{1}$ followed by a copy of $\gamma_{2}.$ If the initial
point is assumed to be the point $-1$ on the first copy of $\gamma_{1}$, then
the change of the argument is $2\pi m.$ If the initial point is the point $-1$
on the second copy of $\gamma_{1}$, then the $m-1$ copies of $\gamma_{1}$
before $\gamma_{2}$ give the winding number $m-1$, but then the curve
$\gamma_{2}$ reverses the orientation so the last copy of $\gamma_{1}$ has
negative orientation with respect to the unit $-i,$ hence the winding number
is $m-2.$ Starting at $-1$ on the third copy of $\gamma_{1}$, would therefore
result in the winding number $m-4$ and so forth.

A few words seem now appropriate, to present a suggestive geometrical
explanation of the reason why the notion of winding number as given in the
case of untwisted loops does not work for the case of twisted loops. Indeed,
consider an untwisted loop $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$

$\gamma(t)=x(t)+\mathcal{I}^{\gamma}(t)y(t)$

If we regard all points $\\{x(t)\\}_{t\in[a,b]}$ as distinct points except the
endpoints, such a $\gamma$ has values in the surface
$S_{\gamma}=\\{x(t)+\mathcal{I}^{\gamma}(t)s:t\in[a,b],s\in\mathbb{R}\\}$;
since $\gamma$ is untwisted, then $\mathcal{I}^{\gamma}:[a,b]\to\mathbb{S}$ is
a loop, and hence it is homotopic to the constant loop
$\mathcal{I}^{\gamma}(0)=\mathcal{I}^{\gamma}(1)$. As a consequence, the
surface $S_{\gamma}$ is homeomorphic to a twodimensional cylinder. Therefore
there is a notion of $\gamma(t)$ being a point of this surface lying on one
side or the other of the “real axis” formed by the points
$\\{x(t)\\}_{t\in[a,b]}$, and hence a notion of winding number with respect to
the origin becomes possible: the situation reduces, naively speaking, to a
planar one. If instead $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ is
twisted, then the path $\mathcal{I}^{\gamma}:[a,b]\to\mathbb{S}$ has antipodal
endpoints, and the surface $S_{\gamma}$ turns out to be homeomorphic to a
Moebius strip. In this last situation, the lack of orientability seems to
exclude the possibility of defining coherently a winding number for the loop
$\gamma$.

##  5\. Obstructions to the existence of lifts of a path

In this section we present sufficient conditions for a path to have a lift, a
companion and to be tame.

As already mentioned, if the path
$\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ misses the real axis, then
the lift to $\mathscr{E}_{\mathbb{K}}^{+}$always exists. On the other hand, if
$\gamma([a,b])\subset\mathbb{R}$, then, necessarily either
$\gamma([a,b])\subset\mathbb{R}^{-}$ and we have the lifts of the form
$\Gamma(t)=\log|\gamma(t)|+I(2k+1)\pi$, or
$\gamma([a,b])\subset\mathbb{R}^{+}$ and then we have the lifts of the form
$\Gamma(t)=\log|\gamma(t)|+I2k\pi$ for any $I\in\mathbb{S}.$ From now on
assume that $\gamma([a,b])$ is not entirely contained in the real axis but it
intersects it.

######  Definition 5.1.

For a path $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ we define the
set $T:=\gamma^{-1}(\mathbb{R})$ to be the obstruction set (for the lift of
$\gamma$) and its points as obstruction parameters.

It is clear that the necessary assumption for a lift of $\gamma$ to
$\mathcal{E}_{\mathbb{K}}^{+}$ to exist is the requirement that $\gamma$ has a
lift on a neighbourhood of every parameter $t,$ in particular, for each $t\in
T.$ It turns out that the existence of local lifts does not necessarily imply
the existence of a global lift; recall that complex curves avoiding $0$ always
have local and global lifts.

In what follows, we start establishing the conditions on the behaviour of
$\gamma$ locally near its obstruction parameters in order to guarantee the
existence first of local lifts and local companions and then of a global lift
and a global companion.

As these conditions depend on the structure of the obstruction set, we start
by considering paths with a finite obstruction set.

######  Definition 5.2.

Let the path $\gamma(t)=x(t)+Y(t):[a,b]\rightarrow\mathbb{H}\setminus\\{0\\}$
be such that $T=\gamma^{-1}(\mathbb{R})=\\{a\leq t_{1}<\ldots<t_{p}\leq b\\}.$
Consider the limits

(5.4) $\lim_{t\rightarrow t_{s}^{\pm}}\frac{Y(t)}{|Y(t)|}.$

Let $t_{s}\in(a,b).$ Then

  * 1)

$\gamma$ is tame at $t_{s}$ if both limits are either equal or opposite. In
particular, if these limits are opposite, then the parameter $t_{s}$ is called
a flip, whereas if they are the same it is called a bounce;

  * 2)

$\gamma$ is semi-tame at $t_{s}$ if it is not tame at $t_{s}$ but both limits
in (5.4) exist;

  * 3)

$\gamma$ is not tame at $t_{s}$ if at least one of the limits in (5.4) does
not exist.

If $t_{s}=a$ (resp. $t_{s}=b$) then the path is tame at $t_{s}$ from the right
(left) if the right (left) limit in (5.4) exists and not tame in all other
cases.

If, in addition, the path $\gamma$ is closed, we adapt the definition of
tameness at the endpoints in the natural way. In particular, $\gamma$ is semi-
tame at $a\simeq b$ if it is tame at $a$ from the right and at $b$ from the
left. If the limits are the same, then $a\simeq b$ is called a bounce and if
they are opposite it is called a flip. In all other cases $\gamma$ is not tame
at $a\simeq b.$

######  Remark 5.3.

A path $\gamma$ in the Example 4.6 (b) does not have the limit (5.4) at $t=0.$

######  Remark 5.4.

The definition of tameness of $\gamma$ at parameters
$\\{t_{1},\ldots,t_{p}\\}$ means precisely that the projectivized imaginary
unit function $[{\mathcal{J}}(\gamma(t))]$ defined on
$[a,b]\setminus\\{t_{1},\ldots,t_{p}\\}$ has a continuous extension to
$\\{t_{1},\ldots,t_{p}\\}$.

The proposition below gives a motivation for the previous definitions.

######  Proposition 5.5.

Let $\gamma(t):[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a path with
finite obstruction set $\gamma^{-1}(\mathbb{R})=\\{a\leq
t_{1}<\ldots<t_{p}\leq b\\}.$ Then $\gamma$ is tame if and only if it is tame
at each $t_{s},s=1,\ldots,p.$

If $\gamma$ is a loop, then it is a tame loop if and only if it is tame at
each $t_{s},s=1,\ldots,p.$

###### Proof.

By assumption, the projectivized imaginary unit function
$[{\mathcal{J}}(\gamma(t))]$ defined on
$[a,b]\setminus\\{t_{1},\ldots,t_{p}\\}$ has a continuous extension to
$[a,b].$ ∎

If $\gamma$ is a tame loop, the lift to $\mathcal{E}_{\mathbb{K}}^{+}$ exists
by Proposition 4.12. However, we want to present also a constructive proof,
because we will use the same techniques to obtain lifts of non tame paths and
to explain the definition of winding number through local data on the
obstruction set.

Without loss of generality we assume that $a\neq t_{1},b\neq t_{p}.$ Consider
the intervals
$I_{0}=[a=:t_{0},t_{1}],I_{1}=[t_{1},t_{2}],\ldots,I_{p}=[t_{p},t_{p+1}=:b]$
and denote the restrictions of $\gamma$ on $I_{s}$ by
$\gamma^{s}:=\gamma|_{I_{s}}.$ The existence of limits (5.4) provides, for any
$s=0,\ldots,p,$ continuous extensions of all functions
$\arg_{k}(\gamma^{s})(t),\mathcal{I}_{k}(\gamma^{s})(t)$ to the endpoints of
$I_{s}.$

Choose an arbitrary $k_{0}\in\mathbb{Z}.$ Setting
$\arg_{k_{0}}(\gamma^{0}(t))=:\arg^{\gamma}(t)$ and
$\mathcal{I}_{k_{0}}(\gamma^{0}(t))=:\mathcal{I}^{\gamma}(t)$ we define
continuous functions $\arg^{\gamma}$ and $\mathcal{I}^{\gamma}$ on
$[a,t_{1}].$ We set
$\textnormal{Arg}^{\gamma}(t):=\arg^{\gamma}(t)\mathcal{I}^{\gamma}(t)$ and
define the lift $\Gamma^{0}:=(\gamma^{0},\textnormal{Arg}^{\gamma^{0}}).$
Consider the endpoint $\gamma(t_{1}).$ If it is a flip, then we define

$k_{1}:=\left\\{\begin{array}[]{ll}k_{0}+1,&\text{if
}\arg_{k_{0}}(\gamma^{0}(t_{1}))=(k_{0}+1)\pi,\\\ k_{0}-1,&\text{if
}\arg_{k_{0}}(\gamma^{0}(t_{1}))=k_{0}\pi.\end{array}\right.$

If it is a bounce then we set $k_{1}:=k_{0}.$ By setting
$\arg^{\gamma}(t):=\arg_{k_{1}}(\gamma^{1}(t))$ and
$\mathcal{I}^{\gamma}(t):=\mathcal{I}_{k_{1}}(\gamma^{1}(t))$ we extend the
functions $\arg^{\gamma}$ and $\mathcal{I}^{\gamma}$ continuously to
$[a,t_{2}].$ We extend the above functions to $[a,b]$ by repeating this
process.

######  Proposition 5.6.

A tame loop $\gamma$ with $\gamma^{-1}(\mathbb{R})=\\{a\leq
t_{1}<\ldots<t_{p}\leq b\\}$ has an even number of flips if and only if
$\gamma$ is untwisted.

###### Proof.

Assume that the loop does not have flips. Then $\mathcal{I}^{\gamma}$ equals
$\mathcal{I}_{k_{0}}\circ\gamma$ for some $k\in\mathbb{Z}$ and so it is
obviously a loop.

For the case of a loop with flips, we assume, without loss of generality, that
$k_{0}=0,$ so we have started with the principal branch and moreover, we also
assume that the parameterization $\gamma:[a,b]\rightarrow\mathbb{H}$ is such
that $\gamma(a)\in\mathbb{R}.$

As in the previous proof, all the functions
$\arg_{0}(\gamma^{s})(t),\mathcal{I}_{0}(\gamma^{s})(t)$ have continuous
extensions to the endpoints of $I_{s}.$

If $a$ is a bounce, then the imaginary unit at $\gamma(b)$ is the same as the
one at $\gamma(a)$, i.e.
$\mathcal{I}_{0}(\gamma^{0})(a)=\mathcal{I}_{0}(\gamma^{p})(b).$ The even
number of flips ensures that the sign of the imaginary unit at the endpoint
remains the same with respect to the one at the principal branch.

If the initial point is a flip, then the imaginary unit function at endpoint
has the opposite sign with respect to the one at the initial point,
$\mathcal{I}_{0}(\gamma^{0})(a)=-\mathcal{I}_{0}(\gamma^{p})(b),$ and to end
up with the same sign there must be an odd number of additional flips
following the first one to ensure that the sign of the unit at the endpoint
remains the same with respect to the one at the principal branch. ∎

The proofs of Propositions 5.5 and 5.6 show that once the lift near the
initial point is chosen, only the flips are relevant for the determination of
the lift near the endpoint; bounces can be discarded. This enables us to
calculate the change of argument and the winding number out of local data at
the intersections of $\gamma$ with the real axis. To determine the change of
the argument we introduce a notion of signature.

######  Definition 5.7.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a given path with
$\gamma^{-1}(\mathbb{R})=\\{a\leq t_{1}<\ldots<t_{p}\leq b\\}$, with points of
$\gamma^{-1}(\mathbb{R})\cap(a,b)$ all tame. Let $a<\xi_{1}<\ldots<\xi_{m}<b$
be those parameters in $\gamma^{-1}(\mathbb{R})$ which are flips. The
signature $\sigma(\gamma)$ is defined by

$\sigma(\gamma):=\sum_{l=1}^{m}\mathrm{sign}(\gamma(\xi_{l}))(-1)^{l}.$

If there are no flips, then we define $\sigma(\gamma):=0.$

The connection between the signature and the change of argument is described
in the following

######  Proposition 5.8.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a tame path with
$\gamma^{-1}(\mathbb{R})=\\{a_{1}\leq t_{1}<\ldots<t_{p}\leq b\\}$ with all
the parameters $\gamma^{-1}(\mathbb{R})\cap(a,b)$ tame. Assume that a lift
$\Gamma$ of $\gamma|_{[a,t_{1}]}$ in $\mathcal{E}_{\mathbb{K}}^{+}$ exists and
equals
$\Gamma(t)=(\gamma(t),\textnormal{Arg}_{k_{0}}^{\gamma}(t))\in\mathcal{E}^{+}_{\mathbb{K}},t\in[a,t_{1}]$
for some $k_{0}\in\mathbb{Z}$. Then the lift of $\gamma|_{[t_{p},b)}$ is given
by
$(\gamma(t),\textnormal{Arg}^{\gamma}_{k_{0}+(-1)^{k_{0}}\sigma(\gamma)}(t)),t\in[t_{p},b).$

######  Remark 5.9.

If $\gamma(b)\in\mathbb{R}$, then a lift of $\gamma$ on $[a,b)$ can be
extended continuously to $b$ if and only if $b$ is tame from the left.

######  Corollary 5.10.

Let $\gamma$ be as in Proposition 5.8 and let
$\gamma(a)=\gamma(b)\not\in\mathbb{R}.$ Then $\sigma(\gamma)$ is even if and
only if $\gamma$ is tame and untwisted. If this is the case, then

$\omega(\gamma)=|\sigma(\gamma)|/2.$

Proof of Proposition 5.8.  Assume that $k_{0}=0,$ so
$\arg(\gamma(a))\in(0,\pi)$ and let the sequence of signs of flips be
alternating starting with $-1,$ i.e. $-1,1,-1,\ldots.$ Then
$\arg^{\gamma}(\gamma(t))$ increases when the path $\gamma$ crosses the real
axis, so $k$ increases by $1$ at each flip (because the sign of the flip
changes); altogether, this occurs
$\sum_{l=1}^{m}(-1)^{l}(-1)^{l}=\sum_{l=1}^{m}\mathrm{sign}(\gamma(\xi_{l}))(-1)^{l}=\sigma(\gamma)$
times. This coincides with winding around the origin of the shadow in the
positive direction. If the sequence of signs starts with $1,$ then
$\arg^{\gamma}(\gamma(t))$ decreases when the path $\gamma$ crosses the
positive real axis and this results in the translation of the interval
$[0,\pi]$ by $\pi\sum_{l=1}^{m}(-1)^{l-1}(-1)^{l}=\pi\sigma(\gamma).$

To prove the general assertion it suffices to show what happens if the
sequence is not alternating at one position.

Assume that we insert in the alternating sequence $-1,1-1\ldots$ the integer
$1$ in the second position, so the sequence is no longer alternating:
$-1,1,1,-1,\ldots.$ This means that we have started from the upper half-plane,
crossed the negative real axis, then the positive real axis with the arguments
in $[2\pi,3\pi].$ Then we have crossed the positive real axis again, hence the
choice of argument at this intersection must be $2\pi.$ Because the point is a
flip, this means that the argument decreases and keeps decreasing till the
end. This is faithfully reflected in the sequence
$s_{l}=(-1)^{l}\mathrm{sign}(\gamma(\xi_{l})),$ because it equals
$1,1,-1,-1,...$ and so the sum
$\sum_{l=1}^{m}\mathrm{sign}(\gamma(\xi_{l}))(-1)^{l}=\sigma(\gamma)$
multiplied by $\pi$ corresponds with the total translation of the initial
interval for the Arg.

Similarly, if we insert $-1$ on the second position, this means that we have
crossed the negative real axis and we have the argument in $[\pi,2\pi]$ but
then we have returned to the negative real axis and in order to have the
argument continuous, at the second crossing the argument $\pi$ must be chosen
and because we have a flip, the argument $\arg^{\gamma}(\gamma(t))$ decreases
and keeps decreasing till the end. The corresponding sequence
$s_{1},s_{2},\ldots$ now equals $1,-1,-1,-1,\ldots$ and
$\sum_{l=1}^{m}s_{l}=\sigma(\gamma).$

The proof for $k_{0}$ even is the same. If $k_{0}$ is odd, this coincides with
considering the conjugate shadow and hence reversed orientation compared, to
$k_{0}$ even, so the signature has to be multiplied by $-\pi$ to get the total
translation of the initial interval for the Arg. $\Box$

In practice this means that once the sequence of $\pm 1$-s is given, we start
by cancelling the pairs of the same numbers until we end up with an
alternating sequence. The number of elements multiplied by minus the first
element is the signature.

If the path $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ is closed,
i.e. $\gamma(a)=\gamma(b)$, then we identify points $a$ and $b$ of $[a,b]$ and
consider the parameterization as
$\gamma:S^{1}\rightarrow\mathbb{K}\setminus\\{0\\},$ so there is no
distinguished initial point. Therefore, in this case we require that for each
$s\in S^{1}$ there exists a neighbourhood of $U_{s}$ of $s$ in $S^{1}$ such
that the lift of $\gamma$ exists on $U_{s}.$

######  Definition 5.11.

Let $\gamma:S^{1}\rightarrow\mathbb{K}\setminus\\{0\\}$ be a continuous loop.
Then a continuous function $\Gamma:i\mathbb{R}\rightarrow\mathcal{E}^{+}$ is a
lift of $\gamma$ if the following diagram commutes:

$\textstyle{i\mathbb{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\exp}$$\scriptstyle{\Gamma}$$\textstyle{\Gamma(i\mathbb{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\subset\mathcal{E}_{\mathbb{K}}^{+}}$$\scriptstyle{{\mathrm{pr}_{1}}\mathrm{}}$$\textstyle{S^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{\gamma(S^{1})\subset\mathbb{K}\setminus\\{0\\}}$

######  Remark 5.12.

A loop with companion always has a (not necessarily closed) lift in the sense
of Definition 5.11. The loop presented in Figure 4 does not have a lift in the
sense of Definition 5.11. The curve is defined by
$\gamma(t)=\cos(t)+i(\sin(t)-t/10)$ for $t\in[0,\pi]$ and
$\gamma(t)=\cos(2\pi-t)-i(2\pi-t)/10+j\sin(2\pi-t)$ for $t\in[\pi,2\pi].$
However, starting from any point with $\gamma(t)\in\mathbb{R}\times
i\mathbb{R}^{+}$ and using the principal branch one can obtain the local lift
of $\gamma$ and prolong it to the interval $[0,2\pi].$

Figure 4. A loop without a lift in the sense of Definition 5.11.

######  Corollary 5.13.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a continuous loop
with $\gamma^{-1}(\mathbb{R})$ nonempty and assume it is not tame at least at
one of the obstruction parameters. Let $a=\xi_{1}<\ldots<\xi_{m}=b\simeq a$
are all the obstruction parameters where $\gamma$ is not tame and assume,
moreover, that $\gamma(\xi_{k})>0.$ Then a lift of $\gamma$ in
$\mathcal{E}^{+}_{\mathbb{K}}$ exists if and only if
$\sigma(\gamma|_{[\xi_{l},\xi_{l+1}]})\in\\{0,-1\\}$ for each
$l=1,\ldots,m-1.$ If it exists, the lift is a loop.

###### Proof.

Because $\gamma$ is not tame at $\xi_{l}$ we can only choose either $k_{0}=0$
or $k_{0}=-1$ and lift the curve in a neighborhood of the point
$\gamma(\xi_{l})$ to $\mathcal{E}^{+}_{\mathbb{K}}$ using the principal branch
of the logarithm. Assume that we have chosen $k_{0}=0$. Then on
$[\xi_{l+1}-\delta,\xi_{l+1})$ for some small $\delta>0$ we may only have
$k=0,-1,$ hence the signature can be either $0$ or $-1$ in order to be able to
extend the lift to $\xi_{l+1}.$ If we have $k_{0}=-1,$ then, since we have to
end up with $k=0,-1$ near $\xi_{i+1},$ the condition is
$(-1)^{k_{0}}\sigma(\gamma|_{[\xi_{l},\xi_{l+1}]})\in\\{0,1\\}$ hence
$\sigma(\gamma|_{[\xi_{l},\xi_{l+1}]})\in\\{0,-1\\}.$ ∎

When we do not have additional information about the set
$\gamma^{-1}(\mathbb{R})$, we must assume that the continuous lift of $\gamma$
exists on a neighbourhood $U$ of $\gamma^{-1}(\mathbb{R})\cap(-\infty,0)$.
This means that the path $\gamma|_{U}$ has a companion, since the restriction
of the function $\arg^{\gamma}$ to $U$ is not vanishing. Recall that, on a
neighbourhood of $\gamma^{-1}(\mathbb{R})\cap(0,\infty)$, the principal branch
of the logarithm is well-defined and hence a lift of $\gamma$ always exists.
This does not imply that a global lift exists.

We now proceed with the detailed description of the possible situations when
$\gamma$ has a companion on a neighborhood of real points and omit the
(trivial) case $\gamma([a,b])\subset\mathbb{R}$.

######  Proposition 5.14.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a path. Then
$\gamma$ has a companion if and only if it has a companion on a neighbourhood
of the obstruction set. The same holds for a loop $\gamma$ with
$\gamma(a)\not\in\mathbb{R}$.

In the sequel we explain how to extend the notion of signature to paths with
infinite obstruction set. Since $\gamma([a,b])$ is compact, there are only
finitely many connected components of $\gamma([a,b])\setminus\mathbb{R}$ with
endpoints of opposite sign.

######  Definition 5.15.

Let $L_{1},\ldots,L_{m}$ be all the connected components of
$\gamma([a,b])\setminus\mathbb{R},$
$L_{l}(t)=\gamma(t),t\in(s_{l},e_{l})\subset[a,b]$ satisfying
$\gamma(s_{l})\gamma(e_{l})<0$ and $a\leq s_{l}<e_{l}\leq s_{l+1}<e_{m}\leq
b$, $l=1,\ldots,m$. We call the components the big arcs and the subdivision
$a\leq s_{l}<e_{l}\leq s_{l+1}<e_{m}\leq b$, $l=1,\ldots,m$ the induced
subdivision. The intervals $[e_{l},s_{l+1}]$ are called obstruction intervals.
If $\gamma$ is closed, then we identify $a$ and $b,$
$e_{0}:=e_{m},s_{m+1}:=s_{1}$ and define also $[e_{0},s_{1}]$ as the
obstruction interval.

Because $\gamma([e_{l},s_{l+1}])$ misses either the positive or the negative
real axis, we define the sign of the obstruction interval as follows.

######  Definition 5.16.

If $\gamma([e_{l},s_{l+1}])\cap(-\infty,0)=\varnothing,$ then
$\mathrm{sign}([e_{l},s_{l+1}])=1$; otherwise, if
$\gamma([e_{l},s_{l+1}])\cap(0,\infty)=\varnothing,$ then
$\mathrm{sign}([e_{l},s_{l+1}])=-1.$

Extend the domains of definition of each $L_{l}$ to its endpoints and let

$\mathcal{I}^{l}(t):=\mathcal{J}(\gamma(t)),t\in(s_{l},e_{l})\mbox{ and
}I_{l}^{s}:=\lim\limits_{t\rightarrow
s_{l}^{-}}\mathcal{J}^{l}(t),I_{l}^{e}:=\lim\limits_{t\rightarrow
e_{l}^{+}}\mathcal{J}^{l}(t)$

be the imaginary units of $L_{l}$ at its endpoints, if the limits exist.

######  Definition 5.17.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a path with
companion $\mathfrak{I}$ with lifts $\pm\mathcal{I}.$ Let $a\leq
s_{1}<e_{1}\leq s_{2}<e_{2}<\ldots\leq s_{m}<e_{m}\leq b$ be the induced
subdivision and $L_{l}$ the big arcs with limits $I_{l}^{e}$ and
$I_{l}^{s},l=1,\ldots,m$.

The interval $[e_{l},s_{l+1}],1\leq l\leq m-1,$ is a bounce with respect to
$\mathfrak{I}$ if $\mathcal{I}$ (or $-\mathcal{I}$) satisfies
$\mathcal{I}(e_{l})=\pm I_{j}^{e},\mathcal{I}(s_{l+1})=\pm I_{l+1}^{s}$ and a
a flip with respect to $\mathfrak{I}$ if $\mathcal{I}$ (or $-\mathcal{I}$)
satisfies $\mathcal{I}(e_{l})=\pm I_{l}^{e},\mathcal{I}(s_{l+1})=\mp
I_{l+1}^{s}.$

######  Remark 5.18.

If $\gamma$ has a companion and $\gamma([e_{l},s_{l+1}])\cap\mathbb{R}$
contains an open set then $\gamma$ always has a companion that makes it a
bounce and a companion that makes it a flip. If the interval $[e_{l},s_{l+1}]$
reduces to a point, then the definition of tameness for intervals coincides
with the definition of tameness for points.

We can now extend the definition of signature also to this general case.

######  Definition 5.19.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a path with the
induced subdivision $a\leq s_{1}<e_{1}\leq s_{2}<e_{2}<\ldots\leq
s_{m}<e_{m}\leq b$ and a companion $\mathfrak{I}.$ Let $1\leq
j_{1}<\ldots<j_{k}\leq m$ be the indices for which the intervals
$[e_{j_{i}},s_{j_{i}+1}]$ are flips. The signature
$\sigma(\gamma,{\mathfrak{I}})$ with respect to the companion $\mathfrak{I}$
is defined as

$\sigma(\gamma,{\mathfrak{I}}):=\sum_{l=1,j_{k}\neq
m}^{k}\mathrm{sign}([e_{j_{l}},s_{j_{l}+1}])(-1)^{l}.$

If $\gamma$ is a loop, then we define the circular signature with respect to
${\mathfrak{I}}$ to be

$\sigma^{c}(\gamma,{\mathfrak{I}}):=\sum_{l=1}^{k}\mathrm{sign}([e_{l},s_{l+1}]))(-1)^{l}.$

If there are no flips, then we define
$\sigma(\gamma,{\mathfrak{I}}):=0,\sigma^{c}(\gamma,{\mathfrak{I}}):=0.$

The following are straightforward generalizations of Proposition 5.8 and
Corollary 5.13

######  Proposition 5.20.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ is a path with the
companion ${\mathfrak{I}}$ and the induced subdivision $a=e_{0}\leq
s_{1}<e_{1}\leq s_{2}<e_{2}<\ldots\leq s_{m}<e_{m}\leq b=s_{m+1}.$ Assume that
a lift $\Gamma$ of $\gamma$ in $\mathcal{E}_{\mathbb{K}}^{+}$ is given by
$\log_{k_{0}},$ $k_{0}\in\mathbb{Z}$ on $[s_{1}-\delta,s_{1}]$ for some
$\delta>0.$ The lift on $[s_{m},e_{m}]$ is given by
$k:=k_{0}+(-1)^{k_{0}}\sigma(\gamma).$

To define the winding number for a closed curve we have to take into account
also the last interval $I_{m}$ and hence consider the closed signature.

######  Corollary 5.21.

Let $\gamma$ be a loop and $\sigma^{c}(\gamma)$ even. Then
$\omega(\gamma,{\mathfrak{I}})=|\sigma^{c}(\gamma,{\mathfrak{I}})|/2.$

######  Corollary 5.22.

Let $\gamma:[a,b]\rightarrow\mathbb{K}\setminus\\{0\\}$ be a loop with the
induced subdivision $a\leq s_{1}<e_{1}\leq s_{2}<e_{2}<\ldots<s_{m}<e_{m}\leq
b.$

Let $1\leq j_{1}<\ldots<j_{k}\leq m$ be the indices for which $\gamma,$
restricted to the neighbourhoods of the intervals
$J_{j_{l}}:=[e_{j_{l}},s_{j_{l+1}}]\subset(0,\infty),$ does not have a
companion and assume $\gamma$ has a companion on a neighbourhood of the
closure of $[a,b]\setminus\cup_{l=1}^{k}J_{j_{l}}.$ Then a lift of $\gamma$ in
$\mathcal{E}_{\mathbb{K}}^{+}$ exists if and only if
$\sigma(\gamma|_{[s_{j_{l}+1},e_{j_{l+1}}]})\in\\{0,-1\\}$ for each
$l=1,\ldots,m-1.$ If it exists, the lift is a loop.

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