Unknotting information from Heegaard Floer homology

Abstract

We use Heegaard Floer homology to obtain bounds on unknotting numbers. This is a generalisation of Ozsváth-Szabó's obstruction to unknotting number one. We determine the unknotting numbers of 9 10 , 9 13 , 9 35 , 9 38 , 10 53 , 10 101 and 10 120 ; this completes the table of unknotting numbers for prime knots with crossing number nine or less. Our obstruction uses a Kirby calculus description of a four-manifold W bounded by the branched double cover of the knot, and a theorem of Cochran and Lickorish which computes the signature of W .

The unknotting number and classical invariants, I

“…Theorem 3.4 is closely related to [Ow08, Theorem 3], which is the main technical theorem of [Ow08]. More precisely, Owens shows in [Ow08, Theorem 3] that if u(K) = u, then Σ(K) can be obtained by Dehn surgery along a u-component link with a certain framing matrix.…”

“…Our understanding of the relation between the n(K) invariant and the presentation matrix for the linking pairing of the double branched cover (cf. Section 3, especially Lemma 3.3) allows us to provide new computable obstructions for u(K) = 2 and u(K) = 3, which are related to Owens' obstruction from [Ow08]. The idea behind the results in Sections 5.2 and 5.3 is the following.…”

“…This fact is used in the original proof of the Lickorish obstruction and it lies at the heart of some of the deepest results on unknotting numbers (see e.g. [OS05] and [Ow08]) which are obtained by studying Heegaard-Floer invariants of the compact 3-manifold Σ(K).…”

“…(1) Lower bounds on the unknotting number have also been obtained using gauge theory [CoL86,KMr93], Khovanov homology [Ras10] and HeegaardFloer homology [Ras03,OS03b,OS05,Ow08,Gr09,Sar10] and various other methods [KM86,Kob89,ST89,Mi98,Tra99,St04,MQ06,GL06]. Note though, that with the exception of the Rasmussen s-invariant, the Ozsváth-Szabo τ -invariant and the Owens obstruction most of the above are in fact obstructions to the unknotting number being equal to one or two.…”

“…(1) Owens [Ow08] shows that the u(K) = 2 obstruction applies to the alternating knots 9 10 , 9 13 , 9 38 , 10 53 , 10 101 , 10 120 which have signature equal to four. On the other hand the algebraic unknotting number equals 2 (using the computer program 'knotorious' we can find explicit algebraic unknotting operations in the sense of [Muk90]).…”