Homological thickness and stability of torus knots

By Marko Stosic. 2005

48 citations recorded by the source corpus.

In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot without changing certain part of its homology, and consequently, we show that there exists stable homology of torus knots conjectured by Dunfield, Gukov and Rasmussen in \cite{dgr}. Since our main tool is the long exact sequence in homology, we have applied our approach in the case of the Khovanov-Rozansky ($sl(n)$) homology, and thus obtained analogous stability properties of $sl(n)$ homology of torus knots, also conjectured in \cite{dgr}.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/476703f62d74cf54a8495313375679c04ad60281

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link (2012, unknown): “We ﬁrst consider the “stable” HOMFLY homology, whose Poincaré polynomial is deﬁned by This limit exists by a theorem of Stoši´c [61].”
Torus knots and the rational DAHA (2012, background): “ m→∞ (a−1q)(n −1)(m 1)H T(n,m). The multiplications by aand qdenote degree shifts which act by the given multiplication when we take the Poincare´ polynomial. This limit exists by a theorem of Stosic [Sto07]. In [DGR06], it was conjectured that HT(n,∞) ∼= Λ∗(ξ 1,...,ξn−1)⊗ C[u1,...,un−1], where ξi h”