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2005•arXiv (Cornell University)Open access

Homological thickness of torus knots

Marko Stošić

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Abstract

In this paper we show that the torus knots Tp,q for 3 ≤ p ≤ q are homologically thick. Even more, we show that we can reduce the number of twists q without changing certain part of homology, and consequently we show that there exists stable homology for torus knots conjectured in [4]. Also, we calculate Khovanov homology groups of low homological degree for torus knots, and we conjecture that the homological width of the torus knot Tp,q is at least p.

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What this paper is about

In this paper we show that the torus knots Tp,q for 3 ≤ p ≤ q are homologically thick. Even more, we show that we can reduce the number of twists q without changing certain part of homology, and consequently we show that there exists stable homology for torus knots conjectured in [4]. Also, we calculate Khovanov homology groups of low homological degree for torus knots, and we conjecture that the homological width of the torus knot Tp,q is at least p.

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Available abstract

In this paper we show that the torus knots Tp,q for 3 ≤ p ≤ q are homologically thick. Even more, we show that we can reduce the number of twists q without changing certain part of homology, and consequently we show that there exists stable homology for torus knots conjectured in [4]. Also, we calculate Khovanov homology groups of low homological degree for torus knots, and we conjecture that the homological width of the torus knot Tp,q is at least p.

Key concepts: Torus, Homology (biology), Khovanov homology, Mathematics, Knot (papermaking), Cellular homology, Diagonal, Combinatorics

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