Torsion in H^2, v(G)-2_A_2(G) and its applications to Khovanov homology of adequate diagrams.

Jozef H. Przytycki

Knots in Washington XXV dedicated to Herbert Seifert on his 100 birthday. Conference on Knot Theory and its Ramifications
December 7-9, 2007
George Washington University
Washington, DC, USA

Organizers
Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Alexander Shumakovitch (GWU), Dan Silver (U. South Al.), Hao Wu (GWU)

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Torsion in H^2, v(G)-2_A_2(G) and its applications to Khovanov homology of adequate diagrams.
by
Jozef H. Przytycki
George Washington University
Coauthors: Radmila Sazdanovic (GWU)

It has been conjectured by Alexander Shumakovitch (announced at Knots in Poland conference in 2003) that any link which is not a connected or disjoint sum of Hopf links and trivial links has a torsion in Khovanov homology. Shumakovitch demonstrated the conjecture for alternating links and Marta Asaeda and myself generalized it for a large class of adequate links (including strongly adequate links). Here we prove the conjecture for those + adequate links D, whose + adequate diagram D has an associated s₊ state graph G_s₊(D) with a cycle of length at least 3. In our work we approximate Khovanov homology of D by chromatic (Helme-Guizon-Rong) cohomology of G_s₊(D). In particular, we prove that for a connected simple graph G of v vertices and cyclomatic number p₁ tor H^{2, v(G)-2}(G) is equal to Z₂^p₁ for G bipartite and Z₂^p₁-1 otherwise.

Date received: December 6, 2007