Seifert circles and knot polynomials

Abstract

In this paper I shall show how certain bounds on the possible diagrams presenting a given oriented knot or link K can be found from its two-variable polynomial PK defined in [3]. The inequalities regarding exponent sum and braid index of possible representations of K by a closed braid which are proved in [5] and [2] follow as a special case.

Braids, transversal links and the Khovanov-Rozansky Theory

“…It's clear that Theorem 1.1 and Corollary 1.3 imply the corresponding results in [4,9]. In particular, the two upper bounds of the self-linking number in Theorem 1.4 are sharper than the bound given by the HOMFLY polynomial, which is implicitly proven in [4,9] (cf.…”

“…In Section 2, we briefly review the definitions of the Khovanov-Rozansky cohomologies and compare the graded Euler characteristics of these cohomologies with the HOMFLY polynomial, which explains why Theorems 1.1, 1.4 and Corollary 1.3 imply the corresponding results by J. Franks, R. Williams and H. Morton in [4,9], including the upper bound of the self-linking number from the HOMFLY polynomial. In Section 3, we study the Khovanov-Rozansky cohomologies of closed braids and prove Theorem 1.1.…”

Bipartite Graphs and Quasipositive Surfaces

“…Moreover, the braid index of K n grows linearly in n, similarly as for twist knots with increasing twist number. This follows, for example, from the inequality of Morton [10] and Franks-Williams [7]. The inequality of Proposition 5.1 implies…”

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.…”