Topology Atlas | Conferences
KNOTS IN WASHINGTON XXI: Skein modules, Khovanov homology and Hochschild homology
December 9-11, 2005
George Washington University
Washington, DC, USA |
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Organizers
Jozef H. Przytycki (GWU), Yongwu Rong (GWU), Alexander Shumakovitch (GWU)
Conference Homepage |
On torsion in the first A3 graph cohomology
by
Milena D. Pabiniak
George Washington University
Coauthors: Jozef H. Przytycki, Radmila Sazdanovic
We will show that for a simple graph G the first
cohomology HA31, 2v(G)−3 (G) behave "nicely" under
one-vertex product (star product), that is:
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HA31, 2v(G1*G2)−3 (G1*G2) = HA31, 2v(G1)−3 (G1) ÅHA31, 2v(G2)−3(G2). |
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This result was motivated by calculations of cohomology
of several small graphs
(e.g. HA31, 7 (P3*P3) = Z3ÅZ3).
Our computations lead us to similar
conjecture (partially proven) for two vertex product. In
particular we proved that if G1**G2 has no "mixed" cycles
of length 3 or 4 then:
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HA31, 2v(G1**G2)−3 (G1**G2) = HA31, 2v(G1)−3 (G1) ÅHA31, 2v(G2)−3(G2). |
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We also propose conjecture:
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Tor(HA31, 2v(G1**G2)−3(G1**G2)) Í Tor(HA31, 2v(G1)−3 (G1))ÅTor(HA31, 2v(G2)−3 (G2)). |
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Additionally we conjecture that every simple graph G containing
K4, complete graph of 4 vertices, will also contain a
Z2 torsion in HA31, 2v(G)−3(G) cohomology
(we have computed that
HA31, 5(K4) = Z33 ÅZ2 ÅZ2 ).
Date received: December 4, 2005
Copyright © 2005 by the author(s).
The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas.
Document # carw-14.