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

Omid Hurson
2018 unpublished
In the first part of the Thesis, we reformulate the Murakami-OhtsukiYamada 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]-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 KhovanovRozansky homology of an oriented link in terms of a combinatorially defined
more » ... 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 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, Carqueville-Murfet, Dolotin-Morozov, Gukov-Iqbal-Kozcaz-Vafa, and Rasmussen.
doi:10.25365/thesis.56502 fatcat:o34canu255a4xeofpmwcxuzdou