Isomorphism of Khovanov chain complexes for a link
Date Issued
2011
Date
2011
Author(s)
Tsai, I-Ming
Abstract
When calculating link invariants, Khovanov has developed a theory which turns the link diagram into a chain complex of direct sum of graded vector spaces defined from the topological quantum field theory (TQFT), then assigns a suitable differential mapping to compute the homology of the chain complex. The result turns out to be related to the Jones polynomial of the link. First we assign each link a new diagram called the graph presentation by replacing the overcrossings of the link by edges, the circles in the Khovaonv chain complexes as vertices. It will help us study the structure of the chain complexes of the link. From Bar Natan’s papers, we use the mapping cobordisms of the chain complex to show that any two link diagrams which are isomorphic by Khovanov chain complexes, then the link diagrams are identical.
Subjects
Khovanov chain complex
Graph presentation
Type
thesis
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