Simplified Khovanov–Rozansky chain complexes of open 2-braids

Abstract

Motivated by the works of Krasner (2009) [8] and Lobb [11], 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.