Colored Khovanov–Rozansky homology for infinite braids

Abstract

We show that the limiting unicolored sl(N ) Khovanov-Rozansky chain complex of any infinite positive braid categorifies a highest-weight projector. This result extends an earlier result of Cautis categorifying highest-weight projectors using the limiting complex of infinite torus braids. Additionally, we show that the results hold in the case of colored HOMFLY-PT Khovanov-Rozansky homology as well. An application of this result is given in finding a partial isomorphism between the HOMFLY-PT homology of any braid positive link and the stable HOMFLY-PT homology of the infinite torus knot as computed by Hogancamp. arXiv:1709.06666v1 [math.QA] 19 Sep 2017) which is an isomorphism for all homological degrees less than y( ).1.1. Outline of paper. In Section 2 we will review the relevant background needed for colored Khovanov-Rozansky sl(N ) complexes of infinite braids, as well as recalling the results of Cautis from [Cau15]. In Section 3 we restate our main results and corollaries and give a detailed proof. In Section 4 we explore the case of colored Khovanov-Rozansky HOMFLY-PT homology and discuss the estimation theorem for braid positive links.

Khovanov–Rozansky homology for infinite multicolored braids

“…We also have results for certain non-color-complete braids (Corollary 5.6). All of these are similar to our previous results in [1,7]. We will also discuss bi-infinite braids, for which defining a coloring and a corresponding limiting complex requires some more care.…”

“…The precise version of Theorem 1.3 will be presented as Corollary 5.10. The proof of Theorem 1.2 effectively generalizes the earlier proofs for different versions of Theorem 1.1 in [1,7]. We present a short outline here.…”

“…The entirety of this section is based on definitions in [14, Section 2.2.2], and expands upon the similar sections in [1,7].…”

“…Queffelec and Rose gave a combinatorial/geometric construction of colored Khovanov-Rozansky homology in terms of 'webs' and 'foams' [10]. It is this construction which we will briefly recall here (again, a large portion of this review is taken nearly verbatim from [1]). Figure 4: Selected relations for sl N -webs (all edges are oriented upwards).…”

“…A natural question for all of these cases one might ask would be what happens if we consider the limiting Khovanov-Rozansky complex of some other infinite braid. Together with Islambouli in [7] and Abel in [1], the author has shown that, for all (semi-)infinite uni-colored braids B that are both positive and complete (a braid is complete if each braid group generator appears infinitely many times; intuitively, this means that B contains all the crossings necessary to build the infinite full twist F ∞ ), the limiting Khovanov-Rozansky complex C N ( B) is chain homotopy equivalent to C N (F ∞ ), the limiting complex of the infinite full twist. Thus we have the following imprecise theorem (see the original papers for the precise versions).…”