Bounds for the Thurston–Bennequin number from Floer homology

Abstract

Using a knot concordance invariant from the Heegaard Floer theory of Ozsváth and Szabó, we obtain new bounds for the ThurstonBennequin and rotation numbers of Legendrian knots in S 3 . We also apply these bounds to calculate the knot concordance invariant for certain knots.

A Concordance Invariant from the Floer Homology of Double Branched Covers

“…(See also [1], [14] for different proofs.) Since T B(K) ≤ 2τ (K) − 1 by the work of Plamenevskaya [28], Rudolph's result is a consequence of ours. Furthermore, Corollary 1.7 can also be applied to knots such as 6 2 , 7 6 , 8 11 and the mirrors of 8 4 , 8 10 , 8 16 , which according to the knot table in [4] have τ > 0 but T B < 0.…”

A family of non-split topologically slice links with arbitrarily large smooth slice genus

“…Our main tool in this paper is the slice-Bennequin inequality (see [Rud95, Rud97, Etn05, AM97, LM98]), which says that for any Legendrian representative K of a knot K, [OS04], and the first inequality is from [Pla04]. Recall that τ is additive under connected sum and insensitive to the orientation of a knot.…”

A criterion for the Legendrian simplicity of the connected sum

“…This result has been improved in many ways, related to classical link invariants such as genus [1,15], polynomials [11,14], or other invariants such as Khovanov homology [12], knot Floer homology [13] and so on. However they are not essential in this paper and we omit the detail.…”