Simplified Khovanov-Rozansky Chain Complexes of Open 2-braids

TL;DR

This paper simplifies the Khovanov-Rozansky chain complexes for open 2-braids and demonstrates how the sl(N) Rasmussen invariant varies linearly with the braid length in certain knots, enabling explicit calculations.

Contribution

It introduces a simplified approach to Khovanov-Rozansky complexes for open 2-braids and derives linear dependence of the Rasmussen invariant on braid length for specific knots.

Findings

Rasmussen invariant depends linearly on braid length in knots with long 2-braids.

Explicit computation of sl(2) Rasmussen invariants for (2,2k+1) cables of slice and amphicheiral knots.

Simplification of chain complexes facilitates invariant calculations.

Abstract

Motivated by the works of Krasner [arXiv:0801.4018] and Lobb [arXiv:1103.1412], we simplify the Khovanov-Rozansky chain complexes of open 2-braids. As an application, we show that, for a knot containing a "long" 2-braid, the sl(N) Rasmussen invariant of this knot depends linearly on the length of this 2-braid. We refine this result for (2,2k+1) cable knots and, as a simple corollary, compute the sl(2) Rasmussen invariants of (2,2k+1) cables of slice and amphicheiral knots.