When is a band-connected sum equal to the connected sum?
Katura Miyazaki





[1]
This question has been posted on
Mathematics Stack Exchange:
https://math.stackexchange.com/questions/2503991/when-is-a-band-connected-sum-equal-to-a-connected-sum-of-knots.


K. Miyazaki]Katura Miyazaki
Faculty of Engineering, Tokyo Denki University, 5 Senju Asahi-cho, Adachi-ku, Tokyo 120–8551, Japan
miyazaki@cck.dendai.ac.jp
[2010]Primary 57M25 Secondary 57M27





We show that
a band-connected sum of knots K_0 and K_1 along a band b
is equal to the connected sum K_0# K_1 if and only if
  b is a trivial band.



When is a band-connected sum equal to the connected sum?^*

    [
    
============================================================





§ INTRODUCTION


Let K_0 ∪ K_1 be a 2-component split link in S^3,
and b: I× I→ S^3 an embedding satisfying
b(I× I) ∩ K_i = b({i}× I ) for i=0,1.
The image b(I× I) is called a band connecting K_0 and K_1;
by abuse of notation, b also denotes b(I× I).
Then the knot K_0∪ K_1 ∪∂ b -int( (K_0 ∪ K_1) ∩ b) is called the band-connected sum of K_0 and K_1 along b, and 
denoted by K_0 #_b K_1,
or K_b for short.

A core of a band b is b(I×{*}), and a cocore of b
is b({*}× I), where *∈intI.
A band b is trivial if there is a splitting sphere for K_0∪ K_1 meeting b in a cocore of b.
Obviously, if b is a trivial band, then K_0 #_b K_1 ≅ K_0#K_1.
We prove  the converse.



If a band-connected sum K_0 #_b K_1 is equal to the connected sum K_0 # K_1, then b is a trivial band.


Regarding the problem when a band-connected sum is composite, 
Eudave-Muñoz proved the following.



If K_b is a composite knot,
then there is a decomposing sphere for K_b which is either
disjoint from b or intersects b in a core of b.


To prove Theorem <ref>
we use Theorem <ref> and the fact that
if K_b ≅ K_0 # K_1, K_b bounds a Seifert surface
of type 1a (Lemma <ref>), which is  defined below.
Any band-connected sum K_b bounds a compact connected surface S containing b.
Such a surface S is called type 1
if S-b is disconnected,
and type 2 otherwise.
In other words,
a type 1 surface for K_0#_b K_1 is a compact connected,
possibly nonorientable, surface that is the union of b and
two connected surfaces bounded by K_i (i=0,1).
A surface S of type 1 is called type 1a if there is 
a splitting sphere for K_0∪ K_1 that is disjoint from S -b,
and type 1b otherwise.

For a composite band-connected sum bounding a surface of type 1a,
we study the configuration of a splitting sphere, a decomposing sphere,
and a type 1a surface,
and obtain Theorem <ref> below.
Theorem <ref> follows from Theorem <ref>.



If K_b is a composite knot and bounds a surface of type 1a,
then there is a decomposing sphere disjoint from b.


As another corollary to Theorem <ref>,
we give sufficient conditions for band-connected sums to be prime.



Assume that K_b = K_0 #_b K_1 bounds
a surface S_0 ∪ b ∪ S_1 of type 1a, where
S_i is a surface bounded by K_i (i=0,1).
Then K_b is prime if b is a nontrivial band
 and either (1) or (2) below holds.


  * K_i is a trivial knot for i=0,1.

  * K_i is prime and S_i is an incompressible Seifert surface for 
i=0,1.





§ PROOFS




If K_b ≅ K_0# K_1, then K_b bounds a Seifert surface of type 1a.


Proof of Lemma <ref>.
The assumption implies g(K_b) = g(K_0) +g(K_1),
where g(·) denotes the genus of a knot.
Then, by <cit.> 
K_b bounds a Seifert surface S of type 1 that is the union of
b and minimal genus Seifert surfaces S_i for K_i (i=0,1).
Let P be a splitting sphere for K_0 ∪ K_1 such that
|P ∩ (S -b)| is minimal among all splitting spheres.
Suppose  P ∩ (S-b) ∅ for a contradiction;
then P∩ (S-b) consists of simple closed curves.
Let Δ_1 be an innermost one
among disks in P bounded by components of P∩(S-b).
Without loss of generality ∂Δ_1 ⊂ S_0.
If ∂Δ_1 is essential in S_0, then
surgery of S_0 along Δ_1 yields a Seifert surface for K_0
with fewer genus than S_0 and possibly a closed surface.
Since S_0 has minimal genus, we see ∂Δ_1 bounds a disk
Δ_2 in S_0.

Each component of Δ_2 ∩ P bounds a disk in Δ_2,
and let Δ_3 (⊂ S_0) be an innermost one among all
such disks in Δ_2;
note Δ_3 ∩ P = ∂Δ_3.
Let B be the 3-ball satisfying ∂ B = P and Δ_3⊂ B,
and B' the closure of a component of B -Δ_3 containing K_0
or K_1.
Shrinking B' slightly, we obtain a splitting sphere ∂ B' with 
|∂ B' ∩ (S-b)| < |P ∩ (S-b)|.
This contradicts the minimality of |P∩ (S-b)|.
It follows P∩ (S-b) =∅, so that S is type 1a
□

Proof of Theorem <ref>.
By Theorem <ref> we may assume that there is a decomposing
sphere Q for K_b= K_0#_b K_1 intersecting b in a core of b.
Take a splitting sphere P for K_0 ∪ K_1,
a decomposing sphere Q for K_b, and a type 1a surface S
bounded by K_b that satisfy conditions (1)—(4) below.



  * P, Q, S are in general position, i.e. any two of these
are in general position, and the intersection of any two of these
and the rest are in general position.


  * P ∩ (S -b) = ∅.


  * Q ∩ b is a core of b, and P∩ b consists of
cocores of b.


  * |Q ∩ S| + |P ∩ Q| + |P ∩ Q ∩ S| is minimal
among all P, Q, S satisfying (1)—(3) above.


We first study the configuration of P ∩ Q and Q ∩ S on Q.
Note that Q ∩ S consists of an arc connecting the two points K_b ∩ Q and possibly simple closed curves.
Let α be the arc component of Q ∩ S, and
set c = Q∩ b, a core of b.
Since c connects  K_0 and K_1, and S -b is disconnected,
c is contained in α.
Note also that P∩ Q, consisting of simple closed curves,
is disjoint from any circle component of Q∩ S
because P∩ (S-b) =∅.
For the same reason P∩ Q intersects α only in c.
See Figure 1.



Assume for a contradiction that
either Q ∩ S or P∩ Q has
a circle component disjoint from α.
Each such component bounds a disk in Q disjoint from α.
Let Δ be an innermost one among all these disks.
If ∂Δ is a component of Q ∩ S,
surger S along Δ to yield a bounded surface S' and
possibly a closed surface.
Since Δ∩ P = ∅,
we see that P ∩ (S' -b) = ∅ and
|Q ∩ S'| < |Q ∩ S|.
This contradicts the minimality assumption (4).

If ∂Δ is a component of P ∩ Q,
surger P along Δ to yield two spheres,
one of which is  a splitting sphere P'.
Since Δ∩ S =∅,
we see that P' ∩ (S -b) = ∅ and
|P' ∩ Q| < |P∩ Q|.
This contradicts the minimality assumption.
We thus obtain the following.



Q∩ S = α, and each component of P∩ Q intersects
α. □


Assume for a contradiction that
there is a component of P∩ Q intersecting α
more than once.
Each such component bounds a disk in Q containing 
at most one endpoint of α.
Let Δ_1 be an innermost one among all such disks.
Then, Δ_1 ∩α consists
of at most one non-properly embedded arc and at least one
properly embedded arc (Figure 2(1)).
Each properly embedded arc of Δ_1 ∩α
and a subarc of ∂Δ_1 cobound a disk in Δ_1
containing no endpoint of α.
Let Δ_2 be an innermost one among all such disks;
note Δ_2 ∩ (S-b) =∅.
We can isotop b along Δ_2 to reduce |P∩ Q∩ S|
(Figure 2(2)), a contradiction to the minimality.
Thus Claim <ref> below is proved.






Each component of P∩ Q intersects α in one point of the core c.
□




Let us take a look at P∩ Q on the splitting sphere P.
Let Δ_3 be an innermost one among all disks in P
bounded by components of P∩ Q; see Figure 3.
Surger the decomposing sphere Q along Δ_3 to obtain
two spheres, at least one of which is a decomposing sphere for K_b.
Let Q' be a decomposing sphere obtained from Q after the surgery.
Without loss of generality Q' is disjoint from K_0
and intersects K_1 in two points.
Let Δ_4 be the closure of the component of b - (b∩ Q')
disjoint from K_0 (Figure 4).
Sliding Q' along the disk Δ_4,
we obtain a decomposing sphere for K_b disjoint from b as claimed.
This completes the proof of Theorem <ref>.
□



Proof of Theorem <ref>.
By Lemma <ref>
K_b bounds a Seifert surface that is the union of minimal genus
Seifert surfaces S_i for K_i (i=0, 1) and b.
We proceed by induction on the product of genera g(K_0)g(K_1).

Case 1. g(K_0)g(K_1) =0.


Since K_0 or K_1 is a trivial knot,
S_0 or S_1 is a disk.
The boundary of a regular neighborhood of the disk
 is a splitting sphere for K_0∪ K_1
intersecting b in a single cocore of b.
It follows that  b is a trivial band.

Case 2. g(K_0)g(K_1)≥ 1.


By Theorem <ref> there is a decomposing sphere Q for K_b
disjoint from b.
Without loss of generality Q intersects K_1 in two points.
Let B be the 3-ball which is bounded by Q and
disjoint from K_0.
Then,  B∩ K_1 is an arc knotted in B.
Let K'_1 be the knot obtained from K_1 by replacing
(B, B∩ K_1) with (D^2× I, {0}× I).
We can regard b as a band connecting K_0 and K'_1.
If b is a trivial band for K_0#_b K'_1,
then it is trivial for K_0#_b K_1.
Since g(K_0)g(K'_1) < g(K_0)g(K_1),
Theorem <ref> is proved inductively.
□

Proof of Proposition <ref>.
Assume for a contradiction that K_b is not prime,
but satisfies either condition (1) or (2) in Proposition <ref>.
If K_b is a trivial knot, then K_0 and K_1 are trivial knots
and especially b is a trivial band (<cit.>),
a contradiction to the assumption of Proposition <ref>.
It follows that K_b is a composite knot.

Then, following the arguments in Case 2 in the proof of
Theorem <ref>,
without loss of generality we can take a 3-ball B such that
B∩ (K_0 ∪ b) =∅ but
 B∩ K_1 is an arc knotted in B.
This implies that K_1 is not a trivial knot,
which violates condition (1).
It follows that K_b satisfies condition (2).

Let K'_1 be K_1 with (B, B∩ K_1) replaced by
(D^2× I, {0}× I).
Since K_1 is prime and B∩ K_1 is a knotted arc in B,
we see K'_1 is a trivial knot.
The Seifert surface S_1 for K_1 intersects ∂ B in
an arc connecting the two points ∂ B ∩ K_1 and
possibly simple closed curves.
However, using the incompressibility of S_1, we can isotop S_1
without meeting S_0∪ b so that S_1∩∂ B is just one arc.
The arc S_1∩∂ B in ∂ B=
∂ (D^2× I) and {0}× I cobound a disk
in D^2× I.
Then, the union S' of the disk and S_1-intB is
an incompressible Seifert surface for the trivial knot K'_1.
It follows that S' and thus S_1-intB are disks.
Isotopy of ∂ B along the disk S_1 -intB
yields a splitting sphere for K_0∪ K_1 intersecting b
in a cocore of b.
This contradicts the assumption that b is a nontrivial band.
□

Gab87
[EM92]EM M. Eudave-Muñoz;
Band sums of links which yield 
composite links. The cabling conjecture for strongly invertible knots, 
Trans. Amer. Math. Soc., 330 (1992), 463–501. 

[Gab87]Ga
D. Gabai, Genus is superadditive under band connected sum, Topology
  26 (1987), 209–210.

[Sch85]Sch2
M. Scharlemann, Smooth spheres in ℝ^4
 with four critical points are standard, 
Invent. Math.  79  (1985),  125–141. 
[Sch87]Sch
M. Scharlemann, Sutured manifolds and generalized Thurston norms, J.
  Differential Geom. 29 (1987), 557–614.
 
