Research ArticleNo Access

Extremal Khovanov homology and the girth of a knot

Radmila Sazdanović

Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA

E-mail Address: rsazdan@ncsu.edu

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and

Daniel Scofield

https://orcid.org/0000-0003-0130-4287

Department of Mathematics, Francis Marion University, Florence, SC 29505, USA

E-mail Address: daniel.scofield@fmarion.edu

Corresponding author.

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https://doi.org/10.1142/S0218216522500833Cited by:1 (Source: Crossref)

Abstract

We show that Khovanov link homology is trivial in a range of gradings and utilize relations between Khovanov and chromatic graph homology to determine extreme Khovanov groups and corresponding coefficients of the Jones polynomial. The extent to which chromatic homology and the chromatic polynomial can be used to compute integral Khovanov homology of a link depends on the maximal girth of its all-positive graphs. In this paper, we define the girth of a link, discuss relations to other knot invariants, and describe possible values for girth. Analyzing girth leads to a description of possible all-A state graphs of any given link; e.g., if a link has a diagram such that the girth of the corresponding all-A graph is equal to $ℓ > 2$ , then the girth of the link is equal to $ℓ .$

Keywords:

AMSC: 57K14, 57K18, 05C31

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