A generators and relations derivation of Khovanov homology of 2 strand braid links

TL;DR

This paper introduces a combinatorial approach to compute Khovanov-Rozansky homology for 2-strand braid links, simplifying calculations and confirming several prior conjectures in link invariants.

Contribution

It reformulates Khovanov-Rozansky homology using combinatorial categories, avoiding complex matrix factorizations and enabling explicit computations for arbitrary crossings.

Findings

Computed invariants for 2-strand braid links with any number of crossings.

Validated and extended previous conjectures on link homologies.

Provided a simplified combinatorial framework for link invariant calculations.

Abstract

In the first part of the Thesis, we reformulate the Murakami-Ohtsuki-Yamada state-sum description of the level n Jones polynomial of an oriented link in terms of a suitable braided monoidal category whose morphisms are Q[q, q-1] s-linear combinations of oriented trivalent planar graphs, and give a corresponding description for the HOMFLY-PT polynomial. In the second part, 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 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 can 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, Carqueville-Murfet, Dolotin-Morozov, Gukov-Iqbal-Kozcaz-Vafa, and Rasmussen