Reducible Legendrian surgeries for \labelsec:main

A proof of Theorem \refthm:main \labelssec:proof-main

Proposition \refprop:decomp guarantees that associated to a reducible Legendrian surgery is a certain Stein filling of a lens space, and consequently a tight contact structure on this lens space. We will prove Theorem \refthm:main by showing that reducible Legendrian surgeries on knots with produce too many tight contact structures in this fashion, appealing to Giroux and Honda’s classification of tight contact structures on (Theorem \refthm:lens-space-classification). We recall our conventions that is -surgery on the unknot and that .

Let be a knot, and suppose that is reducible for some ; write for some . If , then Legendrian surgery on any representative of with and rotation number induces a tight contact structure on with . {proof} By Theorem LABEL:thm:steinfill, the reducible Legendrian surgery gives us a Stein filling of a reducible contact manifold

where bounds a contractible Stein manifold by Proposition LABEL:prop:decomp. Then equation \eqrefeqn:d3-contractible says that , so it remains to compute . By Proposition LABEL:prop:decomp, (since ) and . Thus . In order to compute , we observe that is the class [gompf], where is the cocore of the -handle attached along . Then is generated by a surface of self-intersection , obtained by capping off a Seifert surface for with the core of the -handle, and the map sends . In particular, it sends to , and so

or . We conclude that , as desired. At this point we can give a simple proof of Theorem LABEL:thm:main, which says that if then -surgery on is irreducible for all . {proof}[Proof of Theorem LABEL:thm:main] Suppose that is reducible for some . We know by Proposition LABEL:prop:decomp that and the reducible manifold has a summand of the form with . Since has a Legendrian representative with , and is odd, after possibly reversing the orientation of , it has a representative with and . We can stabilize this representative times with different choices of signs to get representatives with and

and by reversing orientation we also get one with and . Thus the Legendrian representatives of with collectively admit at least different rotation numbers, hence at least values of . For each value of as above, Proposition Document says that admits a tight contact structure with . This value of is uniquely determined by , so the set of rational numbers

(\theequation) |

has at least elements. Now we know from Proposition LABEL:prop:count-xi that has at most tight contact structures. Moreover, by Remark LABEL:rem:xi-conjugate all but at most one of them come in conjugate pairs. Observe that conjugate contact structures have the same invariant, and so the set \eqrefeq:tight-d3-set has at most elements. We conclude that

which is absurd.

## \thesubsection Knots with

In this subsection we discuss the question of which nontrivial knots can have , where is the Seifert genus. We have already shown that reducible surgeries on such knots must have slope at least . Recall from Theorem LABEL:thm:reducible-background that if is also hyperbolic, then Matignon-Sayari showed that reducible surgeries on have slope at most , so then -surgery on cannot be reducible unless .
{proposition}
If is a positive knot, then .
{proof}
Hayden–Sabloff [hayden-sabloff] proved that if is positive then it admits a Lagrangian filling, hence by a theorem of Chantraine [chantraine-concordance] it satisfies , where is the slice genus of , and Rasmussen [rasmussen-s] proved that for positive knots.
It follows from this and item \eqrefi:hyperbolic of Theorem LABEL:thm:reducible-background that if -surgery on a positive knot is reducible, then either is a cable and is the cabling slope, or is hyperbolic and ; thus we have proved Theorem LABEL:thm:pos. (If is also hyperbolic, the claim that there are no essential punctured projective planes in its complement follows exactly as in [hom-lidman-zufelt, Corollary 1.5].) In the case , we can use Heegaard Floer homology to eliminate this last possibility as well.
{theorem}
Positive knots of genus at most 2 satisfy the cabling conjecture.
{proof}
The cabling conjecture is true for genus 1 knots by [boyer-zhang] (see also [matignon-sayari, hom-lidman-zufelt]). If -surgery on the genus 2 positive knot is a counterexample then must be hyperbolic by Theorem LABEL:thm:reducible-background (in particular, is prime) and by Theorem LABEL:thm:pos. As a positive knot of genus 2, is quasi-alternating [jong-kishimoto], hence it has thin knot Floer homology [manolescu-ozsvath]. The signature of is at most , since positive knots satisfy unless they are pretzel knots [przytycki-taniyama, Corollary 1.3] and the cabling conjecture is known for pretzel knots [luft-zhang] (in fact, for all Montesinos knots). Since is a lower bound for the slice genus of , and hence for , we have .
We claim that cannot be fibered. Indeed, Cromwell [cromwell, Corollary 5.1] showed that fibered homogeneous knots have crossing number at most , and since positive knots are homogeneous we need only check the knots with at most 8 crossings in KnotInfo [cha-livingston] to verify that the -torus knot is the only prime, fibered, positive knot of genus , and it is not hyperbolic. Since L-space knots are fibered [ni-fibered], the reducible -surgery on cannot be an L-space. Its lens space summand has order dividing 3, so it must be for some , and if we write for some homology sphere , then it follows from the Künneth formula for [ozsvath-szabo-properties] that is not an L-space.
Since is -thin, the computation of , the plus-flavor of Heegaard Floer homology, was carried out in the proof of [ozsvath-szabo-alternating, Theorem 1.4]; for the claim that the surgery coefficient is “sufficiently large,” see [ozsvath-szabo-knots, Section 4], in particular Corollary 4.2 and Remark 4.3. The result (up to a grading shift in each structure) depends only on the signature and some integers determined by the Alexander polynomial of as follows:
{align*}
\hfp(S^3_3(K), 0) &≅T^+_-5/2 ⊕\ZZ^b_0_(-7/2)

\hfp(S^3_3(K), 1) &≅T^+_-2 ⊕\ZZ^b_1_(-2)
and , where the numbers denote the different structures on and the subscripts on the right denote the grading of either the lowest element of the tower or the summand. The Künneth formula for Heegaard Floer homology implies that each should be isomorphic to as a relatively graded -module (with an absolute shift determined by the correction terms of ), and in particular we must have since is not an L-space. Thus the summands are nontrivial. However, we see that and are not isomorphic as relatively-graded groups, by comparing the gradings of the summand to the gradings of the tower. Thus, the corresponding cannot both be isomorphic to and we conclude that is not reducible after all.
In general the condition can be fairly restrictive, as shown by the following.
{proposition}
If is fibered and , then is strongly quasipositive.
{proof}
Livingston [livingston] and Hedden [hedden-positivity] showed that for fibered knots we have if and only if is strongly quasipositive, where is the Ozsváth–Szabó concordance invariant, which always satisfies [ozsvath-szabo-four-ball]. On the other hand, Plamenevskaya [plamenevskaya-slice-bennequin] proved that for any Legendrian representative of . In particular, if is fibered and not strongly quasipositive then .
It is not true that all fibered, strongly quasipositive knots satisfy . One example is the -cable of the right-handed trefoil : letting , Etnyre and Honda [etnyre-honda-cabling] showed that (which also follows from Corollary LABEL:cor:maxtbcables) but , and since is fibered the latter implies by [hedden-positivity] that it is strongly quasipositive. Etnyre, LaFountain, and Tosun [etnyre-lafountain-tosun] provided many other examples as cables of positive torus knots, but in all such cases we have only if , in which case is not an L-space knot [hom-cabling].
For a hyperbolic example, let be the closure of the strongly quasipositive 3-braid
{align}
β= σ_1^2 σ_2^3 (σ_1 ⋅σ_1 σ_2 σ_1^-1 ⋅σ_2)^3,
cf. [stoimenow, Remark 5.1]. Since is the closure of a 3-braid and its Alexander polynomial

is monic, we know that is fibered [stoimenow, Corollary 4.4] with Seifert genus . {lemma} The knot defined as the closure of \eqrefeq:hyperbolic-braid is hyperbolic. {proof} It suffices to check that is not a satellite, since is not the Alexander polynomial of a torus knot. If is a satellite with companion and pattern , and has winding number in the solid torus, then . Since is fibered, both and are fibered and [hirasawa-murasugi-silver, Theorem 1], and in particular is not constant since it has degree . Since is irreducible, it follows that and , which by inspection implies . Since is fibered with trivial Alexander polynomial, it is unknotted in ; but since it also has winding number 1 it must then be isotopic to the core of the solid torus [hirasawa-murasugi-silver, Corollary 1], and so cannot be a nontrivial satellite. We have the bound where is the Kauffman polynomial of [rudolph-congruence], and so . We note that is not an L-space knot since the coefficients of are not all [ozsvath-szabo-lens], so Conjecture LABEL:conj:maxtblspace remains intact. As evidence for Conjecture LABEL:conj:maxtblspace, we show that it is satisfied by all knots which are known to admit positive lens space surgeries, i.e. the twelve families of Berge knots [berge-conjecture]. (Berge knots are strongly invertible by a result of Osborne [osborne], cf. [watson], so they are already known to satisfy the cabling conjecture [eudave-munoz].) {proposition} If is a Berge knot, reflected if necessary so that it has a positive lens space surgery, then . {proof} Families I–VI are the Berge-Gabai knots [berge-solid-torus, gabai-solid-torus], which are knots in with nontrivial surgeries, and these are known (after possibly reflecting as mentioned above) to be braid positive since they are either torus knots or 1-bridge braids. Families VII and VIII are knots on the fiber surface of a trefoil or figure eight, respectively, and Baker [baker-thesis, Appendix B] showed that they are braid positive as well. Thus in each of these cases the result follows from Proposition Document. The remaining “sporadic” knots, families IX–XII, have because they are all divide knots [yamada-sporadic], hence they satisfy [ishikawa], and as L-space knots they have [ozsvath-szabo-lens], and the result follows. We compile further evidence for Conjecture LABEL:conj:maxtblspace by reducing it to the case of non-cabled knots as follows. We recall that L-space knots are fibered [ni-fibered]. Moreover, torus knots are L-space knots if and only if they are positive, in which case they satisfy by Proposition Document. Finally, the cable of some nontrivial is an L-space knot if and only if is an L-space knot and [hedden-cabling2, hom-cabling], in which case we can apply the following. {proposition} If is fibered and nontrivial and , then whenever . {proof} Because , we assume without loss of generality that ; then the inequality implies , since . Corollary LABEL:cor:maxtbcables then says that , where . Since is fibered, its cable is as well and their Seifert genera are known to be related by , which is equivalent to

But Bennequin’s inequality implies that , so the two must be equal.