Khovanov complexes of rational tangles

Benjamin Thompson

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arXiv · 1701.07525

Khovanov complexes of rational tangles

Benjamin Thompson

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Abstract

We show that the Khovanov complex of a rational tangle has a very simple representative whose backbone of non-zero morphisms forms a zig-zag. Furthermore, this minimal complex can be computed quickly by an inductive algorithm. (For example, we calculate $Kh(8_2)$ by hand.) We find that the bigradings of the subobjects in these minimal complexes can be described by matrix actions, which after a change of basis is the reduced Burau representation of $B_3$.

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Benjamin Thompson. 2017-01-26. Khovanov complexes of rational tangles. https://arxiv.org/abs/1701.07525

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