Graphical methods establishing nontriviality of state cycle Khovanov homology classes

2009/07/02 by Andrew Elliott, Elliott, Andrew · 2 citations
Computer Science · Mathematics · #57M25 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GT #msc:57M25

paper · pdf · doi:10.48550/arxiv.0907.0396

22 pages, 29 figures

arxiv created 2009/07/02 · openalex publication_date 2009/07/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We determine when certain state cycles represent nontrivial Khovanov homology classes by analyzing features of the state graph. Using this method, we are able to produce hyperbolic knots with arbitrarily many diagonals containing nontrivial state cycle homology classes. This gives lower bounds on the Khovanov width of knots whose complexity precludes computation of the full homology.

Citations

• Homology of torus links
• Minimal diagrams of classical and virtual links
• State Cycles, Quasipositive Modification, and Constructing H-thick Knots in Khovanov Homology

Cited by

• A new computational tool for Khovanov cobordism maps
• Equivariant Khovanov-Rozansky Homology and Lee-Gornik Spectral Sequence

Related

• State Cycles, Quasipositive Modification, and Constructing H-thick Knots in Khovanov Homology
• Extremal Khovanov homology and the girth of a knot
• Khovanov homology theories and their applications
• Khovanov homology and the symmetry group of a knot
• Zredukowane homologie Khovanova