Legendrian Hopf links in L(p,1)
Rima Chatterjee Hansjörg Geiges Sinem Onaran

	
R. Chatterjee]Rima Chatterjee
H. Geiges]Hansjörg Geiges
Mathematisches Institut, Universität zu Köln,
Weyertal 86–90, 50931 Köln, Germany
rchatt@math.uni-koeln.de
geiges@math.uni-koeln.de

S. Onaran]Sinem Onaran
Department of Mathematics, Hacettepe University,
06800 Beytepe-Ankara, Turkey
sonaran@hacettepe.edu.tr

R. C. and H. G. are partially supported by the SFB/TRR 191
`Symplectic Structures
in Geometry, Algebra and Dynamics', funded by the DFG
(Project-ID 281071066 - TRR 191); S. O.is partially supported by TÜBİTAK Grant No. 119F411.
R. C. and S. O. would like to thank the Max Planck Institute for
Mathematics in Bonn for its hospitality.








§ ABSTRACT 
We classify Legendrian realisations, up to coarse equivalence,
of the Hopf link in the lens spaces
L(p,1) with any contact structure.


[2020]57K33, 57K10, 57K40, 57R25




Legendrian Hopf links in L(p,1)
    [
    
===============================









§ INTRODUCTION



By the (positive) Hopf link L_0⊔ L_1 in the lens space
L(p,1) we mean the (ordered, oriented) link
formed by the two rational unknots given by the spines of the genus 1
Heegaard decomposition, oriented in such a way that their
rational linking number equals 1/p. By the discussion in
<cit.>, this characterisation determines the link up to
isotopy and a simultaneous change of orientations (which can be effected
by an orientation-preserving diffeomorphism of L(p,1));
the key result in the background is the uniqueness of the
genus 1 Heegaard splitting up to isotopy.

The lens space L(p,1) with its natural orientation as a quotient
of S^3 can be realised by a single (-p)-surgery on an unknot. The Hopf link
in this surgery diagram is shown in Figure <ref>;
the link is positive when both L_0 and L_1 are oriented as meridians of
the surgery curve in the same way. Indeed, if we label meridian and longitude
(given by the Seifert framing) of the surgery curve by μ and λ,
the (-p)-surgery amounts to replacing a tubular neighbourhood of
the surgery curve by a solid torus V_1=S^1× D^2, with meridian
μ_1={*}×∂ D^2 and longitude λ_1=S^1×{1}
glued as follows:

    μ_1⟼ pμ-λ,   λ_1⟼μ.

Then L_1=μ=λ_1 may be thought of as the spine of V_1,
and L_0 as the spine of the complementary solid torus V_0,
with meridian μ_0=λ and longitude λ_0=μ.

A Seifert surface Σ_0
for pL_0 is made up of the radial surface in V_0 between
the p-fold covered spine pL_0 and the curve

    pλ_0-μ_0=pμ-λ=μ_1,

and a (positively oriented) meridional disc in V_1. The spine L_1
intersects this disc positively in a single point.



In this paper we extend the classification of Legendrian Hopf links
in S^3, see <cit.>, to Legendrian Hopf links in L(p,1)
for any p∈, with any contact structure. For a brief survey
of the rather scant known results on the classification of Legendrian links
(with at least two components) we refer to <cit.>.
Our result is the first classification of Legendrian links in a
3-manifold other than S^3.

Legendrian links in overtwisted contact manifolds are either
loose
(the link complement is still overtwisted) or exceptional
(the link complement is tight). In the exceptional case, the
link complement may or may not contain Giroux torsion.
In contrast with <cit.>, we only consider the case of vanishing
Giroux torsion (what in <cit.> we called
strongly exceptional);
the case of Giroux torsion adds numerical complexity
but no significant insight. Beware that the individual components
of an exceptional Legendrian link may well be loose.

Our classification of the Legendrian realisations
(up to coarse equivalence, i.e. up to a contactomorphism of the
ambient manifold) of the Hopf link
in L(p,1) is in terms of the rational classical invariants as defined
in <cit.>.


The rational Thurston–Bennequin invariant of the L_i, in any contact
structure on L(p,1), is of the form _(L_i)=_i+1/p
with _i∈.



By symmetry it suffices to show this for L_0, where we can use
the above description of a Seifert surface Σ_0 for pL_0.
The contact framing of L_0 is given by a curve _0μ_0+λ_0
with _0∈. With the identifications in the surgery
description of L(p,1) we have

    _0μ_0+λ_0=_0λ+μ=
    _0(pλ_1-μ_1)+λ_1.

We can push this curve on ∂ V_1 a little into V_1,
and then the intersection number with the meridional disc bounded by
μ_1 (as part of Σ_0) equals _0p+1. To obtain
_(L_0), this number has to be divided by p.


The rational rotation
number _ is well defined (i.e. independent of a choice
of rational Seifert surface), since the Euler class of any
contact structure on L(p,1) is a torsion class.

We also include the d_3-invariant of
the overtwisted contact structures. This invariant does not play a direct
role in the classification, but in some cases we need it to determine
whether the link components are loose or exceptional, when we appeal to
the classification of exceptional Legendrian rational unknots in L(p,1)
achieved in <cit.>. The notation ξ_d stands
for an overtwisted contact structure with d_3=d. For a complete homotopical
classification of the overtwisted contact structures one also needs
to know their Euler class (or, in the presence of 2-torsion, a finer
d_2-invariant). The Euler class can be computed from
the surgery diagrams we present, using the recipe of
<cit.>. These computations, which are rather involved, are omitted here.
In the cases where we use contact cuts to find the Legendrian realisations,
the homotopical classification of the contact structure in question is
more straightforward.

We use the notation  for any one of the tight contact
structures on L(p,1). The homotopical data of the relevant
contact structures containing Legendrian realisations of the Hopf
link can easily be read off from Figure <ref>. These tight
structures are Stein fillable and hence have zero Giroux torsion.

The essence of the following main theorem is that the classical invariants
suffice to classify the Legendrian realisations of the Hopf link in
L(p,1), so the Hopf link is what is called Legendrian simple.



Up to coarse equivalence, the Legendrian realisations of the
Hopf link in L(p,1), p≥ 2, with zero Giroux torsion in
the complement, are as follows. In all cases, the classical
invariants determine the Legendrian realisation.


  (a) In (L(p,1),) there is a unique realisation for any
combination of classical invariants (_(L_i),_(L_i))=
(_i+1/p,_i-/p),
i=0,1, in the range _0,_1<0 and

    ∈{-p+2,-p+4,…,p-4,p-2},


    _i∈{_i+1,_i+3,…,-_i-3,-_i-1}.

For fixed values of _0,_1<0 this gives a total
of _0_1(p-1) realisations.

  (b) For _0=0 and _1≤ 0 there are
|_1-1| exceptional realisations, all living in
(L(p,1),ξ_d) with d=(3-p)/4, made up of an exceptional
component L_0 with classical invariants (_(L_0),_(L_0))=
(1/p, 0), and a loose component L_1 with
invariants (_(L_1),_(L_1))=(_1+1/p,_1), where

    _1∈{_1,_1+2,…,-_1-2,-_1}.


  (c) For _0=0 and _1>0 the exceptional
realisations are as follows; in all cases both components are loose.


  (c1) For _1=1 there are two realisations,
with classical invariants

    (_(L_0),_(L_0))=(1/p,±2/p)

and

    (_(L_1),_(L_1))=(1+1/p,
    ±(1+2/p)).

They live in (L(p,1),ξ_d) with d=(3p-p^2-4)/4p.

  (c2) For _1=2, there are three exceptional
realisations. Two of them live in (L(p,1),ξ_d),
where d=3p-p^2-4/4p, and have classical invariants
(_(L_0),_(L_0)) as in (c1)
and

    (_(L_1),_(L_1))=(2+1/p,±(2+2/p)
    )).

The third one lives in  (L(p,1),ξ_d) with d=7-p/4,
and the invariants are (_(L_0),_(L_0))=(1/p,0)
and (_(L_1),_(L_1))=(2+1/p,0).			

  (c3) For _1>2, there are four exceptional
realisations. The classical invariants are listed in Table
<ref> in Section <ref>.
	

  (d) For _0,_1>0 the exceptional
realisations, with both components loose, are as follows:


  (d1) For _0=_1=1, there are exactly p+3
exceptional realisations. They live in (L(p,1),ξ_d),
where d=7p-^2/4p. They have classical invariants

    (_(L_0),_(L_0))=
    (1+1/p,/p)

and

    (_ℚ(L_1),_ℚ(L_1))
    =(1+1/p,/p),

where

    ∈{-p-2,-p,…, p,p+2}.


  (d2) For _0=1 and _1>1, there are 2(p+2) exceptional
realisations, whose classical invariants are given in Table <ref>.

  (d3)For _0>1 and _1>1, there are 4(p+1) exceptional
realisations, whose classical invariants are given in
Table <ref>.
	

  (e) For _0<0 and _1>0 the exceptional
realisations are as follows. Here L_0 is loose; L_1 is exceptional.


  (e1) For _1=1, there are exactly
|_0|(p+1) exceptional realisations. They live in
(L(p,1),ξ_d) where d=(3p-^2)/4p, where

    ∈{-p, -p+2,…, p-2, p}.

The classical invariants are

    (_(L_0),_(L_0))=
    (_0+1/p,_0-/p)

and

    (_ℚ(L_1),_ℚ(L_1))=
    (1+1/p,-/p),

where 

    _0∈{_0+1,_0+3,…, -_0-3, -_0-1}.


  (e2) For _0<0 and _1>1, there are 2|_0|p exceptional
realisations, whose classical invariants are given in
Section <ref>.

		


The proof of Theorem <ref> largely follows the strategy
used in <cit.> for Legendrian Hopf links in S^3:
find an upper bound on the number of exceptional realisations by enumerating
the tight contact structures on the link complement, and then
show that this bound is attained by giving explicit realisations.

Most of these explicit realisations are in terms
of surgery diagrams, but as in <cit.> there is a
case where a surgery presentation eludes us, and we have to use
contact cuts instead. This case (c1), which is being treated in
Section <ref>, contains most of the conceptually novel aspects
in the present paper. In contrast with <cit.>, we no longer
have a global frame for the contact structure; therefore, the computation
of rotation numbers requires the explicit description of rational
Seifert surfaces, and frames over them, in the context of topological cuts.
The discussion in Section <ref> should prove
useful in a more general analysis of the contact topology of lens
spaces via contact cuts.


One has to be a little careful when comparing this result with
the classification of Legendrian Hopf links in S^3 in <cit.>.
For instance, in case (c1), the contact cut description we use
in Section <ref>
corresponds for p=1 to the interpretation of S^3 as lens space
L(1,1), whereas in <cit.> we read S^3 as L(1,0). In case (a),
the surgery diagram for S^3 is empty, and the discussion in the present
paper only makes sense for p≥ 2. In most other cases, however, one
obtains the correct results for S^3 by allowing p=1 in
Theorem <ref>.






§ UPPER BOUND FOR EXCEPTIONAL REALISATIONS


In this section we determine the number of tight contact structures on the
complement of a Legendrian Hopf link L_0⊔ L_1 in
L(p,1), in terms of the
Thurston–Bennequin invariant of the link components.
We start with S^3, decomposed into
two solid tori V_0,V_1 forming a Hopf link (in the traditional
sense), and a thickened torus T^2×[0,1], i.e.
    S^3=V_0∪_∂ V_0=T^2×{0} T^2× [0,1]
    ∪_T^2×{1}=∂ V_1 V_1.

Write μ_i,λ_i for meridian and longitude
on ∂ V_i, and define the gluing in the decomposition above by

    μ_0         =     S^1×{*}×{0},
    λ_0     =    {*}× S^1×{0},
    μ_1         =    {*}× S^1×{1},
    λ_1     =     S^1×{*}×{1}.
 

As in the introduction, we think of the Hopf link in L(p,1)
as being obtained by (-p)-surgery along the spine of V_1.
Slightly changing the notation from the introduction, we write
μ_1',λ_1' for meridian and longitude of the solid torus
V_1' reglued in place of V_1, so that the gluing
prescription becomes

    μ_1'⟼ pμ_1-λ_1,   λ_1'⟼μ_1.


Given a Legendrian Hopf link L_0⊔ L_1 in L(p,1)
with _(L_i)=_i+1/p, we can choose V_0,V_1'
(sic!) as standard neighbourhoods of L_0,L_1, respectively.
This means that ∂ V_0 is a convex surface with two dividing
curves of slope 1/_0 with respect to (μ_0,λ_0);
the slope of ∂ V_1' is 1/_1 with respect to
(μ_1',λ_1').

Now, on T^2×[0,1] we measure slopes on the T^2-factor with
respect to (μ_0,λ_0).
So we are dealing with a contact structure
on T^2×[0,1] with convex boundary, two dividing curves
on either boundary component, of slope s_0=1/_0 on
T^2×{0}, and of slope s_1=-p-1/_1 on T^2×{1}, since

    _1μ_1'+λ_1'=_1(pλ_0-μ_0)+λ_0=
    (_1p+1)λ_0-_1μ_0.

Recall that a contact structure on T^2×[0,1]
with these boundary conditions is called minimally twisting
if every convex torus parallel to the boundary has slope between s_1
and s_0.

The following proposition covers all possible pairs (_0,_1),
possibly after exchanging the roles of L_0 and L_1.



Up to an isotopy fixing the boundary, the number N=N(_0,_1)
of tight, minimally twisting contact structures
on T^2×[0,1] with convex boundary, two dividing curves
on either boundary component of slope s_0=1/_0 and
s_1=-p-1/_1, respectively, is as follows.


  (a) If _0,_1<0, we have N=_0_1(p-1).

  (b) If _0=0 and _1≤ 0, then N=|_1-1|.

  (c) If _0=0, _1≥ 1:


  (c1) N(0,1)=2.

  (c2) N(0,2)=3.

  (c3) For all _1>2, we have N(0,_1)=4.


  (d) If _0,_1>0:


  (d1) N(1,1)=p+3.

  (d2) For all _1>1, we have N(1,_1)=2(p+2).

  (d3) For all _0,_1>1, we have N=4(p+1).
	

  (e) If _0<0, _1>0:


  (e1) For all _0<0 and _1>1, we have
N=2|_0|p.

  (e2) For all _0<0, we have
N(_0,1)=|_0|(p+1).





So that we can use the classification of tight contact structures on
T^2× [0,1] due to Giroux <cit.> and Honda <cit.>,
we normalise the slopes by applying
an element of Diff^+(T^2)≅(2,) to T^2×[0,1]
such that the slope on T^2×{0} becomes s_0'=-1,
and on T^2×{1} we have s_1'≤ -1. If s_1'<-1,
the number N is found from a continuous fraction expansion

    s_1'=r_0-1r_1-1r_2-⋯-1r_k
    =:[r_0,…,r_k]

with all r_i<-1 as

    N=|(r_0+1)⋯(r_k-1+1)r_k|,

see <cit.>.
The vector [ x; y ] stands for
the curve xμ_0+yλ_0, with slope y/x.





 §.§ Case (a)


We have 

    [     0    -1;     1 -_0+1 ][ _0;  1 ]
    =[ -1;  1 ]

and 

    [     0    -1;     1 -_0+1 ][   -_1; p_1+1 ]
    =[             -p_1-1; -_1-p_0_1+p_1-_0+1 ],

which implies

    s_1'=_0-1+_1/p_1+1=
    _0-1-1-p-1_1.

For _1<-1 we read this as [_0-1,-p,_1]; for
_1=-1 and p>2, as [_0-1,-p+1]; for _1=-1
and p=2, as [_0].
In all three cases this gives  N=|_0_1(p-1)|.





 §.§ Case (b)


For _0=0 and _1≤ 0 we use the transformation

    [  -p  -1; p+1   1 ][ 0; 1 ]
    =[ -1;  1 ]

and

    [  -p  -1; p+1   1 ][   -_1; p_1+1 ]
    =[   -1; 1-_1 ].

This gives s_1'=-1+_1, whence N=|_1-1|.





 §.§ Case (c)
 For t_0=0 and t_1≥ 2 we have

    [ -(p+1)     -1;    p+2      1 ][ 0; 1 ]
    =[ -1;  1 ]

and

    [ -(p+1)     -1;    p+2      1 ][   -_1; p_1+1 ]
    =[   _1-1; -2_1+1 ].

A continued fraction expansion for s_1'=(-2_1+1)/(_1-1) is given by

    [-3,-2,-2,…, -2__1-2].

Hence, for _1>2 we get 
N=|(-2)(-1)⋯ (-1)(-2)|=4; for _1=2 we have N=3.

For _1=1 we work instead with the transformation

    [  -2p   -1; 1+2p    1 ][ 0; 1 ]
    =[ -1;  1 ]

and

    [  -2p   -1; 1+2p    1 ][  -1; p+1 ]
    =[ p-1;  -p ],

which gives

    s_1'=-p/p-1=
    [-2,-2,…, -2_p-1],

whence N=2.





 §.§ Case (d)


For _0>0 and _1>0 we compute

    [        -p    -1+p_0;       1+p 1-(1+p)_0 ][ _0;  1 ]
    =[ -1;  1 ]

and 

    [        -p    -1+p_0;       1+p 1-(1+p)_0 ][   -_1; p_1+1 ]
    =[            -1+p_0+p^2_0_1; 1-_0-_1-p_0-p_0_1-p^2_0_1 ].

Hence,

    s_1'=-1-_0+_1+p_0_1/-1+p_0+p^2_0_1.


For _0=1, the continued fraction expansion is

    s_1'=[-2,-2,…, -2_p-1, -(p+3),
    -2,-2,…, -2__1-1].

Thus, N=p+3 for _1=1, and N=2(p+2) for _1>1.

For _0,_1>1 we have the continued fraction expansion

    s_1'=[-2,-2,… ,-2_p-1, -3,
    -2,-2,…, -2__0-2, -(p+2),
    -2, -2,…, -2__1-1],

whence N=4(p+1).





 §.§ Case (e)


For _0<0 and _1>0 we use the transformation

    [    -1  _0-1;     2 1-2_0 ][ _0;  1 ]
    =[ -1;  1 ]

and

    [    -1  _0-1;     2 1-2_0 ][   -_1; p_1+1 ]
    =[   -1+_0+_1-p_1+p_0_1; 1-2_0-2_1+p_1-2p_0_1 ].

Then

    s_1'=-2-1+p_1/-1+_0+_1-p_1+p_0_1
    =[-2, _0-1,-(p+1),-2,-2,…,-2__1-1].

For _1>1 this yields N=2|_0|p; for
_1=1 we get N=|_0|(p+1).






§ COMPUTING THE INVARIANTS FROM SURGERY DIAGRAMS



Except for the case (c1) discussed in terms of contact cuts in
Section <ref>, we are going to describe the
Legendrian realisations of the Hopf link in L(p,1)
as front projections of a Legendrian link in a contact surgery
diagram for (L(p,1),ξ) involving only contact (± 1)-surgeries.
Here we briefly recall how to
compute the classical invariants from such a presentation;
for more details see <cit.>.

Write M for the linking matrix of the surgery diagram,
with the surgery knots K_1,…,K_n given auxiliary orientations,
and (K_j,K_j) equal to the topological surgery framing.
The extended linking matrix of a Legendrian knot L_i in this
surgery presentation is

    M_i=([         0 (L_i,K_1)         ⋯ (L_i,K_n); (L_i,K_1)                              ;         ⋮                   M          ; (L_i,K_n)                               ]).






 §.§ Thurston–Bennequin invariant


Write _i for the Thurston–Bennequin invariant of L_i
as a Legendrian knot in (S^3,), that is, before performing
the contact surgeries. Then, in the surgered contact manifold, one has

    _(L_i)=_i+ M_i/ M.






 §.§ Rotation number


Write _i for the rotation number of L_i before the surgery.
With

    :=((K_1),…,(K_n))

and

    _i:=((L_i,K_1),…,(L_i,K_n))

we have

    _(L_i)=_i-⟨,M^-1_i⟩.






 §.§ The d_3-invariant


The surgery diagram describes a 4-dimensional handlebody X with
signature σ and Euler characteristic χ=1+n. Let
c∈ H^2(X) be the cohomology class determined by c(Σ_j)=
(K_j), where Σ_j is the oriented surface made up
of a Seifert surface for K_j and the core disc of the
corresponding handle. Write q for the number of contact (+1)-surgeries.

Then the d_3-invariant is given by the formula

    d_3(ξ)=1/4(c^2-3σ-2χ)+q,

where c^2 is computed as follows: find the solution vector 
of the equation M=; then c^2=^M=⟨,⟩.

The signature σ can be computed from the linking matrix
corresponding to the surgery diagram; more efficiently, one can usually
compute it using Kirby moves as described in <cit.>.





§ HOPF LINKS IN TIGHT L(P,1)


For given values of _0,_1<0, we have
_0_1(p-1) explicit realisations in (L(p,1),) as
shown in Figure <ref>. Here the numbers k,k_i and
ℓ,ℓ_i refer to the exterior cusps, so that
the surgery curve K has =-k-ℓ+1 (and _i=-k_i-ℓ_i+1),
and to obtain L(p,1)
by a contact (-1)-surgery on K we need k+ℓ=p. The
_i can take any negative value. With L_0, L_1 both oriented
clockwise, the Hopf link is positive,
and in this case we have _i=ℓ_i-k_i.



The values of the classical invariants as claimed in
Theorem <ref> (a) now follow easily from the formulas in
Section <ref>, with _i=_i, _i=_i,
and =(K). For the d_3-invariant we observe
that σ=-1, χ=2, and c^2=-^2/p, whence

    d_3=-1/4(1+^2/p).


There are no realisations in (L(p,1),) with one of the
_i being non-negative, since Legendrian rational unknots
in (L(p,1),) satisfy _i<0 by
<cit.> or <cit.>.

This proves part (a) of Theorem <ref>.



The values of the d_3-invariant found above exhausts
all possibilities for the d_3-invariant of the tight contact structures
on L(p,1); see <cit.>.






§ LEGENDRIAN HOPF LINKS VIA CONTACT CUTS



In this section we use contact cuts to find Legendrian realisations
of Hopf links in L(p,1) for the case (c1).





 §.§ L(p,1) as a cut manifold


We first want to give a topological description of L(p,1) as a
cut manifold in the sense of Lerman <cit.>. We start
from T^2×[0,1]=S^1× S^1×[0,1] with coordinates
x,y∈ S^1=/ and z∈[0,1]. Collapsing the first S^1-factor
in S^1× S^1×{0} is equivalent to attaching a solid torus
to  T^2×[0,1] along T^2×{0} by sending the
meridian of the solid torus to S^1×{*}×{0}. This, of course,
simply amounts to attaching a collar to a solid torus, and the meridian
of this `fattened' torus is μ_0:=S^1×{*}×{1}. As longitude
we take the curve λ_0:={*}× S^1×{1}.

Now, as described in Section <ref>,
L(p,1) is obtained from this
solid torus by attaching another solid torus, whose meridian is glued
to the curve pλ_0-μ_0. Equivalently, we may collapse
the foliation of T^2×{1} by circles in the class pλ_0-μ_0.

Consider the p-fold cover (/)×(/p)× [0,1]→
(/)×(/)×[0,1]. Set λ̃_0:={*}×
(/p)×{1}. The foliation of (/)×(/p)× [0,1]
by circles in the class λ̃_0-μ_0 descends to the
foliation defined by pλ_0-μ_0. This exhibits L(p,1)
as a _p-quotient of L(1,1)=S^3. Beware that in <cit.>
we used the description of S^3 as L(1,0) in the cut construction,
so for p=1 one has to take this into account when comparing
the discussion here with the results in <cit.>.





 §.§ Contact structures from contact cuts


A contact structure on T^2×[0,1] will descend to the cut manifold
L(p,1) if, at least near the boundary, it is invariant under the
S^1-action whose orbits on the boundary are collapsed to a point,
and if the S^1-action is tangent to the contact structure along the
boundary.

We define a∈ (0,π/2) by the condition tan a=p. For
ℓ∈_0, consider the contact form

    α_ℓ:=sin((a+ℓπ)z)  x+
    cos((a+ℓπ)z)  y

on T^2×[0,1]. This 1-form is invariant under the flows of both
∂_x and ∂_y. Along T^2×{0}, we have
∂_x∈α_ℓ; along T^2×{1},
the vector field -∂_x+p∂_y is in α_ℓ.
Thus, α_ℓ descends to a contact form on L(p,1).

By <cit.>, adapted to the description of S^3
as L(1,1), the lift of α_0 to S^3 is the standard tight contact
structure . Indeed, the map (x,y)↦ (x+y,y) sends this lifted
contact structure on T^2× [0,1], up to isotopy rel boundary and
compatibly with the cuts on the boundary, to the one shown in <cit.>
to define  on S^3=L(1,0).

This means that
α_0 is the unique (up to diffeomorphism)
universally tight contact structure on L(p,1). The contact structure
α_ℓ+1 is obtained from α_ℓ by
a π-Lutz twist. In particular, α_2 is obtained from
α_0 by a full Lutz twist, and hence is homotopically
equivalent to the universally tight contact structure;
see <cit.>. The surgery description for
the universally tight contact structure on L(p,1) is a contact
(-1)-surgery on a (p-2)-fold stabilised standard Legendrian
unknot (with =-1 and =0) in (S^3,), all stabilisations
having the same sign. This contact structure has d_3-invariant equal
to (3p-p^2-4)/4p and Euler class ±(p-2) (in terms of the
natural cohomology generator); see <cit.>
and <cit.>.





 §.§ A Legendrian Hopf link in (L(p,1),α_2)


The contact planes in α_2 on T^2×[0,1] have
slope 0 with respect to (∂_x,∂_y) at z=0.
As z increases to z=1, the contact planes twist (with decreasing slope)
for a little over 2π, until they reach slope -p for the third time
(and after passing through slope ±∞ twice), similar to
Figure 18 in <cit.>.

Define z_0<z_1 in the interval [0,1] by the conditions

    a+2π z_0=π/2   and   
    -sin((a+2π)z_1)+(p+1)cos((a+2π)z_1)=0.

This means that at z=z_0 the contact planes are vertical for the
first time, and T^2×{z_1} is the second torus (as z increases
from 0 to 1) where the characteristic foliation has slope -(p+1).

Consider the link L_0⊔ L_1 made up of a (0,1)-curve
on the torus {z=z_0} and a (-1,p+1)-curve on {z=z_1}.
This link is topologically isotopic to the one made up of
a (0,1)-curve on {z=0} and a (-1,p+1)-curve on {z=1}.
Since these respective curves have intersection number ± 1 with
the curves we collapse at either end,

    (0,1)∙ (1,0)=-1,    (-1,p+1)∙ (-1,p)=1,

they are isotopic to the spine of the solid tori attached at either end,
and so they constitute a Hopf link.

A Seifert surface Σ_0 for pL_0, regarded as p times the spine
of the solid torus attached at z=0, is given by the following pieces:


  - a helicoidal annulus between pL_0 and a (-1,p)-curve on {z=0},

  - an annulus between this (-1,p)-curve on {z=0} and the same
curve on {z=1},

  - the meridional disc attached to the latter curve at z=1.

Then

    L_1∙Σ_0=(-1,p+1)∙(-1,p)=-p + (p+1)=1.

Notice that ∂_y is positively transverse to Σ_0
on T^2×[0,1], so our calculation gives the correct sign
of the intersection number. Thus, L_0⊔ L_1 is a positive
Hopf link.

Both components L_0 and L_1 of this Legendrian Hopf link are
loose, since the contact planes twist by more than π in either interval
(z_0,1) and (0,z_1), so the cut produces an overtwisted disc.


The Legendrian Hopf link L_0⊔ L_1 in (L(p,1),α_2)
is exceptional.



Arguing by contradiction, suppose there were an overtwisted disc in the
complement of the link. This disc would persist in the complement of the
transverse link L_0'⊔ L_1' obtained by pushing L_0 a little
towards z=0, and L_1 towards z=1.
In T^2×[0,1], this link is transversely isotopic
to the link made up of a (0,1)-curve on {z=0} and a (-1,p+1)-curve
on {z=1}. In the cut manifold L(p,1), this gives a transverse isotopy
(and hence an ambient contact isotopy)
from L_0'⊔ L_1' to the transverse link made up of the collapsed
boundaries (T^2×{0})/S^1 and (T^2×{1})/S^1.
The complement of the latter, however, is
contactomorphic to (T^2× (0,1),α_2), which is tight, as it
embeds into the standard tight contact structure on ^3.






 §.§ Computing _


A Legendrian push-off of L_0 is simply a parallel (0,1)-curve
L_0' on { z=z_0}. The topological isotopy from pL_0 to p times
the spine of the solid torus attached at z=0 can be performed in the
complement of L_0'. So the rational Thurston–Bennequin invariant
(as defined in <cit.>) of L_0 is given by

    _(L_0)=1/pL_0'∙Σ_0=
    1/p(0,1)∙(-1,p)=1/p.


For L_1 we argue similarly. A Legendrian push-off L_1' is given by a
parallel (-1,p+1)-curve on {z=z_1}. The isotopy of L_1 to the spine
of the solid torus attached at z=1 can be performed in the complement
of L_1', so _(L_1) can be computed as L_1'∙Σ_1,
where Σ_1 is a Seifert surface for p times that spine,
made up of the following pieces:


  - a helicoidal annulus between pL_1 and a curve on
{z=1} in the class

    p(-1,p+1)-(p+1)(-1,p)=(1,0)

(see below for an explanation of this choice),

  - an annulus between this (1,0)-curve on {z=1} and the same
curve on {z=0},

  - the meridional disc attached to the latter curve at z=0.

For the first constituent of this Seifert surface, notice that the p-fold
covered spine of a solid torus can be joined by an annulus to any
simple curve on the boundary in a class pλ+kμ, k∈,
where λ is any longitude on the boundary, and μ the meridian.
Or, more directly, simply observe that in our case
μ=(-1,p), and (1,0)∙ (-1,p)=p.

The vector field ∂_y is positively transverse to Σ_1
on T^2× [0,1], so we obtain

    _(L_1)=1/pL_1'∙Σ_1=1/p(p+1)
    =1+1/p.






 §.§ Frames for α_ℓ


A frame for α_ℓ on T^2×[0,1], compatible with the
orientation defined by α_ℓ, is given by

    ∂_z   and   
    X_ℓ:=cos((a+ℓπ)z)∂_x-
    sin((a+ℓπ)z)∂_y.

This frame does not descend to a frame of the contact structure
on L(p,1).

At z=0 we have X_ℓ=∂_x. If we think of the cut at
z=0 as being defined by attaching a solid torus, with meridian
being sent to the x-curves, the vector field ∂_z is
outward radial along the boundary of the solid torus, and X_ℓ=∂_x
is positively tangent to the meridional curves. It follows that a frame
of α_ℓ that extends over the cut at z=0 is given by

    cos(2π x)∂_z-sin(2π x)X_ℓ   and   sin(2π x)∂_z+cos(2π x)X_ℓ;

cf. <cit.>.

At z=1, where we collapse the flow lines of -∂_x+p∂_y, we
have

    X_ℓ=±(cos a ∂_x-sin a ∂_y)=
    ∓cos a  (-∂_x+p ∂_y),

depending on ℓ  being even or odd.
If we think of the cut again as attaching a solid torus, now ∂_z
is inward radial along the boundary of the solid torus, and X_ℓ
is tangent to the meridional curve (positively for ℓ odd,
negatively for ℓ even). So the frame that extends over the cut
at z=1 is

    cos(2π x)∂_z+sin(2π x)X_ℓ   and   
    -sin(2π x)∂_z+cos(2π x) X_ℓ   for ℓ even,

and

    cos(2π x)∂_z-sin(2π x)X_ℓ   and   sin(2π x)∂_z+cos(2π x) X_ℓ   for ℓ odd.

Thus, only for ℓ odd do we have a global frame.





 §.§ Computing _


We now look again at the Legendrian Hopf link L_0⊔ L_1
in (L(p,1),α_2).

For the computation of the intersection number L_1∙Σ_0
between L_1 and a Seifert surface for pL_0 we had the freedom
to isotope L_0 (topologically) in the complement of
L_1 to the spine of the solid torus
attached at z=0. When we want to compute _(L_0), we need to
work with a Seifert surface for the original Legendrian L_0.
This means that we need to work with the Seifert surface Σ_0
made up of the following pieces:


  - a p-fold covered straight annulus between pL_0, i.e.the (0,p)-curve on the torus {z=z_0}, and the p-fold covered spine
of the solid torus attached at z=0,

  - a helicoidal annulus between p times the spine
and a (-1,p)-curve on the torus {z=0},

  - an annulus between this (-1,p)-curve on {z=0} and the same
curve on {z=1},

  - the meridional disc attached to the latter curve at z=1.



This surface is not embedded, but the computation of rotation numbers
is homological, so this is not a problem.


Our aim is to find a frame of α_2 defined over Σ_0.
We begin with the frame defined over T^2×[0,1]
by cos(2π x)∂_z+sin(2π x)X_2
(and its companion defining the correct orientation), which is the
one that extends over the cut at z=1. In particular, this frame
is defined over the third and fourth constituent of Σ_0, and we need
to extend it over the first two pieces of Σ_0.

Write (/)× D^2 with coordinates (y;r,θ) for the solid torus
attached at z=0. We pass to the p-fold covers

    (/)^2× [0,1]⟶ (/)^2×[0,1],  
    (x,y,z)⟼ (x,py,z)

and

    (/)× D^2⟶ (/)× D^2,  
    (y;r,θ)⟼ (py;r,θ),

where the lifted pieces of Σ_0 are embedded:
a straight annulus between the (0,1)-curve on {z=z_0}
and the spine of the solid torus, plus a helicoidal annulus between
the spine and the (-1,1)-curve on the boundary of the solid torus.
For the following homotopical considerations, we may think
of α_2 as being extended over that solid torus
as the constant horizontal plane field.

Along the (-1,1) curve on {z=0}, parametrised as /∋ t↦
(x(t),y(t),0)=(-t,t,0), the frame we are considering is

    cos(2π t)∂_z-sin(2π t)∂_x.

In the cylindrical coordinates (y;r,θ) on the solid torus
this translates into the frame

    cos(2π t)∂_r-sin(2π t)∂_θ

along the curve t↦ (y(t),θ(t))=(t,-2π t) on {r=1}.

Next, we translate this into Cartesian coordinates (u,v) on
the D^2-factor. With r∂_r=u∂_u+v∂_v
and ∂_θ=u∂_v-v∂_u, and the curve in question
being (u(t),v(t))=(cos 2π t,-sin 2π t), this gives the frame

    [ cos(2π t)(cos(2π t)∂_u-sin(2π t)∂_v)-;  sin(2π t)(cos(2π t)∂_v+sin(2π t)∂_u)                                     =            cos(4π t)∂_u-sin(4π t)∂_v. ]


This formula defines the extension of the frame over the helicoidal annulus
and the part of the straight annulus inside the solid torus.
The intersection of the straight annulus with the torus {z=0}
is the (0,1)-curve, parametrised as t↦ (x(t),y(t))=(0,t),
and if we take this curve to be given by { u=1,v=0} (so that
∂_u=∂_z and ∂_v=∂_x), the frame is now
written as

    cos(4π t)∂_z-sin(4π t)∂_x,

which extends over the annulus between the (0,1)-curve on {z=0}
and that on {z=z_0} (i.e. the p-fold cover of L_0) as

    cos(4π t)∂_z-sin(4π t)X_2.

Notice that at z=z_0 we have X_2=-∂_y. The
orientation of α_2 is defined by (∂_z,X_2), so
the frame makes two negative rotations with respect to the
tangent vector ∂_y=-X_2 of the p-fold
covered L_0. We conclude that _(L_0)=2/p.




Next we compute _(L_1) in an analogous fashion. We now use the frame
cos(2π x)∂_z-sin(2π x)X_2 on T^2× [0,1], which is the
one that extends over the cut at z=0. The cut we perform at z=1
corresponds to the attaching of a solid torus (/)× D^2
using the gluing map

    μ={0}×∂ D^2⟼ (-1,p)   and   λ=(/)×{1}⟼ (-1,p+1).

Note that with respect to the orientation defined by (∂_x,∂_y),
the intersection number of meridian and longitude is

    μ∙λ=(-1,p)∙(-1,p+1)=-1,

which is what we want, since ∂_z is the inward normal
of the solid torus along its boundary, so this boundary is oriented
by (∂_y,∂_x). Notice also that the tangent direction
of μ coincides with -X_2.

The Legendrian knot L_1 on {z=z_1} is a (-1,p+1)-curve, so the
parallel curve on {z=1} is in the class of λ. The relevant parts
of the Seifert surface Σ_1 for pL_1 in the p-fold cover, i.e.the lift with respect to the map

    (/)× D^2⟶ (/)× D^2,  
    (s;r,θ)⟼ (ps;r,θ),

are the following:


  - a straight annulus between the lifted longitude and the spine,

  - a helicoidal annulus between the spine and the (1,0)-curve on
{z=1}; recall that pλ-(p+1)μ=(1,0).


At z=1, the frame cos(2π x)∂_z-sin(2π x)X_2
we have chosen equals

    cos(2π x)∂_z+sin(2π x)cos a(-∂_x+p∂_y).

Along the (1,0)-curve, parametrised as t↦ (x(t),y(t))=(t,0),
this frame is

    cos(2π t)∂_z+sin(2π t)cos a(-∂_x+p∂_y)=
    -cos(2π t)∂_r+sin(2π t)∂_θ.

Translated into Cartesian coordinates (u,v) on the D^2-factor of
the solid torus, the curve (or rather its projection to
the D^2-factor) becomes

    (u(t),v(t))=(cos 2π(p+1)t,-sin 2π(p+1)t),

and translating the frame into Cartesian coordinates as above, we obtain

    [ -cos(2π t)(cos(2π(p+1)t)∂_u-
    sin(2π(p+1)t)∂_v)+;   sin(2π t)(cos(2π(p+1)t)∂_v+
    sin(2π(p+1)t)∂_u)                                               =;             -cos(2π(p+2)t)∂_u+sin(2π(p+2)t)∂_v.                                                 ]

This formula defines the extension of the frame over the helicoidal
annulus and the straight annulus inside the solid torus.
Along the intersection of the straight annulus with
the boundary of the solid torus, given again by {u=1,v=0},
and further on the annulus between pλ=p(-1,p+1) and pL_1,
this frame extends as

    cos(2π(p+2)t)∂_z-sin(2π(p+2)t)X_2,

since at (u,v)=(1,0) the vector ∂_u is the
outward normal -∂_z, and ∂_v points in
meridional direction, which is identified with -X_2.

This frame makes p+2 negative rotations with respect
to the tangent direction X_2 of L_1, which yields
_(L_1)=(p+2)/p=1+2/p.




Since we may flip the orientations of L_0 and L_1 simultaneously,
this gives us in total two realisations, with invariants

    (_(L_0),_(L_0))=(1/p,±2/p),
       
    (_(L_1),_(L_1))=(1+1/p,
    ±(1+2/p)).



For p=1, this accords with the case (1,± 2) and (2,± 3)
discussed in <cit.>.






§ DETECTING EXCEPTIONAL LINKS



Before we turn to the Legendrian realisations of Legendrian
Hopf links in L(p,1) in terms of contact surgery diagrams, we discuss
how to establish that a given Hopf link is exceptional, and how to decide
whether the individual components are loose or exceptional.

First one needs to verify that the contact structure given
by the surgery diagram is overtwisted.
If the d_3-invariant differs from that of any of the tight
structure (see Remark <ref>), this
is obvious. If the d_3-invariant does match that of a tight
structure, overtwistedness can be shown by exhibiting
a Legendrian knot in the surgered manifold that violates the
Bennequin inequality <cit.> for Legendrian knots
in tight contact 3-manifolds. In all our examples, one of L_0 or L_1
will have this property. Alternatively, one can appeal to
the classification of Legendrian rational unknots in L(p,1) with
a tight contact structure <cit.>. If the invariants
of L_0 or L_1 do not match those listed there (in particular,
_(L)=+1/p with  a negative integer),
then the contact structure
must be overtwisted. Again, this covers all cases (b)–(e).

Secondly, we need to establish that the contact structure on the
link complement L(p,1)∖(L_0⊔ L_1) is tight. The method we use is
to perform contact surgeries on L_0 and L_1, perhaps also on
Legendrian push-offs of these knots, such that the
resulting contact manifold is tight. If there had been an overtwisted
disc in the complement of L_0⊔ L_1, this would persist
after the surgery.

Here we rely on the cancellation lemma from <cit.>,
cf. <cit.>, which says that a contact
(-1)-surgery and a contact (+1)-surgery along a Legendrian knot
and its Legendrian push-off, respectively, cancel each other.
For instance, if by contact (-1)-surgeries on L_0 and L_1
we can cancel all contact (+1)-surgeries in the surgery diagram,
and thus obtain a Stein fillable and hence tight contact 3-manifold,
the Legendrian Hopf link will have been exceptional.

To determine whether one of the link components is loose,
we sometimes rely on the classification of exceptional rational unknots
in L(p,1) given in <cit.>:



Up to coarse equivalence, the exceptional rational unknots in
L(p,1) are classified by their classical invariants
_ and _. The possible values of
_ℚ are +1/p with ∈_0. For =0, there is a
single exceptional knot, with _=0. For =1, there are p+1
exceptional knots, with

    _∈{-1, -1+2/p, -1+4/p, …,
    -1+2p/p}.

For ≥ 2, there are 2p exceptional knots, with

    _∈{±(-2+2/p),
    ±(-2+4/p),…,
    ±(-2+2p/p)}.



In a few cases, the invariants of one link component
equal those realised by an exceptional rational unknot,
but we detect looseness by computing the d_3-invariant
and observing, again comparing with <cit.>,
that it does not match that of an exceptional realisation.





§ EXCEPTIONAL HOPF LINKS



In this section we find exceptional Legendrian realisations of the
Hopf link, except case (c1)—which is covered by
Section <ref>—, in contact surgery diagrams for L(p,1).
This completes the proof of Theorem <ref>.





 §.§ Kirby diagrams


We begin with some examples of Kirby diagrams of the Hopf link
that will be relevant in several cases of this classification.
The proof of the following lemma is given by the Kirby moves in the
corresponding diagrams.


(i) The oriented link L_0⊔ L_1 in the surgery diagram
shown in the first line of Figure <ref>
is a positive or negative Hopf link in L(p,1), depending
on k being even or odd.
The same is true for the links shown
in the first line of Figure <ref> and <ref>,
respectively. 

(ii) The oriented link L_0⊔ L_1 shown in Figure <ref>
is a positive Hopf link in L(p,1).

(iii) The oriented link L_0⊔ L_1 in the first line
of Figure <ref>
is a positive or negative Hopf link depending on k_0 and k_1
having the same parity or not.



















 §.§ Legendrian realisations
 



  §.§.§ Case (b)


The |_1-1| realisations with _0=0 and _1≤ 0
are shown in Figure <ref>. Here k and ℓ denote the
number of stabilisations, with k+ℓ=-_1, so that
_1=_1-1.
 


The linking matrix M of the surgery diagram is the
((p+1)× (p+1))-matrix

    [  0 -1 -1  ⋯ -1; -1  0 -1  ⋯ -1; -1 -1  0  ⋯ -1;  ⋮  ⋮  ⋮  ⋱  ⋮; -1 -1 -1  ⋯  0 ].

The determinant of this matrix is M=-p.
The extended linking matrix for L_i, i=0,1, is the
((p+2)× (p+2))-matrix built in the same fashion as M, so that
(M_i) =-p-1.

It follows that 

    _(L_0)=-1+p+1/p=1/p

and

    _(L_1)=_1-1+p+1/p=_1+1/p.

Since =0, we get
_(L_0)=0 and

    _(L_1)∈{_1, _1+2,…,-_1-2,-_1}.

 
For the calculation of d_3, we observe that c^2=0, χ=p+2,
and σ=p-1. Thus, 

    d_3=1/4(0-2(p+2)-3(p-1))+p+1=1/4(-5p-1)+p+1=3-p/4.

 
By Theorem <ref>, there are no exceptional realisations
of a rational unknot in L(p,1) with _ equal to that of L_1,
so L_1 is loose. The component L_0 is exceptional, as can
be seen by performing surgery on it. A contact (-1/p+2)-surgery
on L_0 has the same effect as taking p+2 Legendrian push-offs of L_0
and doing a (-1)-surgery on each of them. This cancels the
(+1)-surgeries in the diagram and hence produces a tight contact
3-manifold.





  §.§.§ Case (c2)


In this case we have exactly three Legendrian realisations.
The left-hand side of Figure <ref>, where the
rational rotation numbers of L_0 and L_1 will be shown to be
non-zero, gives two realisations (one with L_0,L_1 oriented as shown,
the second with orientations flipped simultaneously.
The right-hand side, where the rotation numbers are
zero, gives the third one.



We begin with the left-hand side.
The linking matrix for the surgery diagram, ordering the surgery knots
from top to bottom, and with all surgery knots oriented clockwise,
is the ((p+3)× (p+3))-matrix

    M=[ -2 -1 -1  ⋯ -1; -1  0 -1  ⋯ -1; -1 -1  0  ⋯ -1;  ⋮  ⋮  ⋮  ⋱  ⋮; -1 -1 -1  ⋯  0 ],

with M=p.
 
The extended linking matrices for L_0 and L_1 are

    M_0= ([  0 -1 -1  ⋯ -1; -1            ; -1     M      ;  ⋮            ; -1             ]),

with M_0=1+p, and

    M_1= ([ 0 3 1 ⋯ 1; 3        ; 1   M    ; ⋮        ; 1         ]),

with M_1=1+5p. This yields

    _(L_0)=-1+1+p/p=1/p   and   _(L_1)=-3+1+5p/p=2+1/p,
 
so we are indeed in the case (_0,_1)=(0,2).
Further, 

    =(2,0,0,…, 0_p+2),  _0=(-1,-1,…, -1_p+2),  _1= (3,1,…,1_p+2).

A simple calculation gives

    M^-1_0=(-1/p,1/p,…,1/p)   and   
    M^-1_1=(-(2p+1)/p,1/p,…,1/p).

Hence

    _(L_0)=0-2·-1/p=2/p   and   _(L_1)=-2+2·2p+1/p=2+2/p.

The second realisation is given by reversing the orientations of
both L_0 and L_1; this changes the sign of the _(L_i)
(as is best seen by also changing the orientations of the surgery curves,
although the choice of orientation here does not matter).

For the diagram on the right, the calculation of _(L_i)
does not change, but now we have =0 and _1=0,
whence _(L_0)=_(L_1)=0. This yields the third realisation.

For the calculation of the d_3-invariant we observe that both diagrams
in Figure <ref> give σ=p-1 and χ=p+4.
The diagram on the right has =0, which yields
d_3=(7-p)/4. For the diagram on the left, 
the solution  of M= is =(-2-2/p,2/p,…,2/p)^.
Then c^2=⟨,⟩=-2(2+2/p) and d_3=
(3p-p^2-4)/4p.

In either case, the contact structure is overtwisted by the observations in
Section <ref>.
A contact (-1)-surgery along L_1 and a contact (-1/p+2)-surgery
along L_0 cancels all (+1)-surgeries, so the link is exceptional.
Both components are loose, since the classical invariants do not
match those of Theorem <ref>.





  §.§.§ Case (c3)


Two of the four Legendrian realisations in this case are shown in
Figure <ref>, with L_0 and L_1 both oriented
clockwise or counter-clockwise for k even, L_1 having the opposite
orientation of L_0 for k odd; cf. Figure <ref>.
The other two are obtained by
flipping the shark with surgery coefficient -1.
The invariants are shown in
Table <ref>. We omit the calculations of the invariants
in this and the remaining cases. The arithmetic is lengthy but
not inspiring; the detailed calculations are available from
the authors upon request.






The contact structure is overtwisted, because we have rational unknots
with _i≥ 0. A contact (-1)-surgery on L_1 and a contact
(-1/p+2)-surgery on L_0 will cancel all contact (+1)-surgeries,
so the link is exceptional.

Except for (_0,_(L_0))=(0,0),
the invariants of L_0 and L_1 are not realised by
exceptional rational unknots, see Theorem <ref> or,
for a better overview, <cit.>; hence these knots are loose.
So is L_0 in the first line of Table <ref>, since
the value of the d_3-invariant does not match that of an exceptional
realisation; again see <cit.>. This is the first
instance where we need the d_3-invariant to detect looseness.





  §.§.§ Case (d1)


The p+3 Legendrian realisations with invariants as listed
in Theorem <ref> are shown in Figure <ref>.
The surgery knot of which L_1 is a push-off is a (p+2)-fold
stabilisation of the standard Legendrian unknot (compare with
the topological surgery framing in Figure <ref>);
placing the stabilisations left or right gives the p+3 choices.
Flipping the orientations of both L_0 and L_1 has the same
effect as exchanging the number of stabilisations on the left-
and right-hand side, so this does not give any new choices.



The argument for the link being exceptional is as in the preceding case.
According to <cit.>,
the classical invariants are realised by exceptional rational unknots,
but only in the overtwisted contact structure with d_3=(3p-^2)/4p,
so the link components here are loose.





  §.§.§ Case (d2)


The 2(p+2) Legendrian realisations are shown in Figure <ref>,
with L_0,L_1 given the same orientation for k even, the opposite, for
k odd. The factor 2 comes from simultaneously exchanging
the orientations of L_0 and L_1.
The factor p+2 comes from the placement of p+1
stabilisations on the surgery knot near the bottom of the diagram
(compare with Figure <ref>). We write

    ∈{-p-1,-p+1,…, p-1,p+1}

for the p+2 possible rotation numbers of this surgery knot.
The invariants are listed in Table <ref>.
The remaining arguments are as before.










  §.§.§ Case (d3)


The 4(p+1) Legendrian realisations are shown in Figure <ref>.
For k_0,k_1 of equal parity, we give L_0 and L_1 the same
orientation; for k_0,k_1 of opposite parity, the opposite
orientation. A factor 2 comes from exchanging the orientations
simultaneously, a factor 2 from the two diagrams, and a factor p+1
from placing p stabilisations left or right on the surgery knot
at the centre of each diagram, cf. Figure <ref>.
The rotation number of this surgery knot is denoted by

    ∈{ -p, -p+2,…, p-2,p}.

The invariants are shown in Table <ref>.
As before, suitable contact surgeries along L_0,L_1
and comparison with <cit.> show that the link
is exceptional and the components loose.










  §.§.§ Case (e1)


Figure <ref> shows |_0|(p+1) Legendrian realisations.
Placing the |_0|-1 stabilisations left or right gives |_0| choices.
The surgery knot is the standard Legendrian unknot with p
stabilisations, which gives p+1 choices. Flipping the orientations
of L_0,L_1 simultaneously is the same as flipping the whole
diagram (and the left/right stabilisations), so this does not
give any additional realisations. Write

    ∈{-p,-p+2,…, p-2,p}

for the rotation number of the surgery knot. A straightforward calculation
then gives _0=_0 (and hence the right number
of realisations) and the other classical invariants as listed
in Theorem <ref>, with _0:=_0.
The d_3-invariant takes the value (3p-^2)/4p.
Since _1>0, the contact structure is overtwisted.
A contact (-1)-surgery on L_1 gives (S^3,), so L_1 is
exceptional. On the other hand, L_0 must be loose, since _0<0.







  §.§.§ Case (e2)



The 2|_0|p realisations for this case are shown in
Figure <ref>, with L_0 and L_1 both oriented
clockwise or counter-clockwise for k odd, L_1 having the opposite
orientation of L_0 for k even; cf. Figure <ref>.
Here L_0 is an unknot with
_0<0 (which will again turn out to equal _0);
this gives us |_0| choices distinguished by

    _0:=_0∈{_0+1,_0+3,…,-_0-3,-_0-1}.

The topmost surgery knot has Thurston–Bennequin invariant -p;
this gives us p choices distinguished by the rotation number

    ∈{-p+1,-p+3,…,p-3,p-1}.

The factor 2 in the number of choices comes from simultaneously
flipping the orientations of L_0 and L_1.

One then computes that

    _(L_0)=±(_0++1/p),   _(L_1)=±(_1-1++1/p),

and

    d_3=1/4(3+2-^2-1/p).

The argument for showing that L_0 is loose, and L_1 exceptional,
is as in case (e1).






10

baet12
K. Baker and J. Etnyre,
Rational linking and contact geometry,
Perspectives in Analysis, Geometry, and Topology,
Progr. Math. 296
(Birkhäuser Verlag, Basel, 2012), 19–37.

chke
R. Chatterjee and M. Kegel,
Contact surgery numbers of Σ(2,3,11) and L(4m+3,4),
.

dige04
F. Ding and H. Geiges,
A Legendrian surgery presentation of contact 3-manifolds,
Math. Proc. Cambridge Philos. Soc.
136 (2004), 583–598.

dgs04
F. Ding, H. Geiges and A. I. Stipsicz,
Surgery diagrams for contact 3-manifolds,
Turkish J. Math.
28 (2004), 41–74.

geig08
H. Geiges,
An Introduction to Contact Topology,
Cambridge Stud. Adv. Math. 109
(Cambridge University Press, Cambridge, 2008).

geon15
H. Geiges and S. Onaran,
Legendrian rational unknots in lens spaces,
J. Symplectic Geom.
13 (2015), 17–50.

geon20
H. Geiges and S. Onaran,
Legendrian Hopf links,
Quart. J. Math. 71 (2020), 1419–1459.

giro00
E. Giroux,
Structures de contact en dimension trois et bifurcations des
feuilletages de surfaces,
Invent. Math.
141 (2000), 615–689.

hond00I
K. Honda,
On the classification of tight contact structures I,
Geom. Topol.
4 (2000), 309–368;
erratum: Geom. Topol.
5 (2001), 925–938.

lerm01
E. Lerman,
Contact cuts,
Israel J. Math.
124 (2001), 77–92.



