Computing the Khovanov homology of 2 strand braid links via generators and relations

TL;DR

This paper introduces a combinatorial approach to compute the Khovanov-Rozansky homology of 2-strand braid links, simplifying calculations and confirming several previous conjectures.

Contribution

It extends existing polynomial invariants to a combinatorial framework, enabling efficient computation of Khovanov-Rozansky homology for 2-strand braids with arbitrary crossings.

Findings

Successfully computed the Khovanov-Rozansky invariant for 2-strand braid links with any number of crossings.

Confirmed and extended previous theoretical predictions and conjectures.

Provided a simplified, combinatorial method avoiding complex matrix factorizations.

Abstract

In "Homfly polynomial via an invariant of colored plane graphs", Murakami, Ohtsuki, and Yamada provide a state-sum description of the level $n$ Jones polynomial of an oriented link in terms of a suitable braided monoidal category whose morphisms are $\mathbb{Q}[q,q^{-1}]$-linear combinations of oriented trivalent planar graphs, and give a corresponding description for the HOMFLY-PT polynomial. We extend this construction and express the Khovanov-Rozansky homology of an oriented link in terms of a combinatorially defined category whose morphisms are equivalence classes of formal complexes of (formal direct sums of shifted) oriented trivalent plane graphs. By working combinatorially, one avoids many of the computational difficulties involved in the matrix factorization computations of the original Khovanov-Rozansky formulation: one systematically uses combinatorial relations satisfied by these matrix factorizations to simplify the computation at a level that is easily handled. By using this technique, we are able to provide a computation of the level $n$ Khovanov-Rozansky invariant of the 2-strand braid link with $k$ crossings, for arbitrary $n$ and $k$, confirming and extending previous results and conjectural predictions by Anokhina-Morozov, Beliakova-Putyra-Wehrli, Carqueville-Murfet, Dolotin-Morozov, Gukov-Iqbal-Kozcaz-Vafa, Nizami-Munir-Sohail-Usman, and Rasmussen.