The Gysin sequence and the sl(N) homology of T(2,m)

TL;DR

This paper demonstrates how the Gysin sequence can be used to establish an isomorphism between the sl(N) homology of T(2,m) torus knots and the cohomology of associated SU(N) representation spaces, avoiding explicit calculations.

Contribution

It introduces a novel application of the Gysin sequence to prove isomorphisms in knot homology and representation space cohomology without direct computation.

Findings

Gysin sequence effectively relates knot homology and representation space cohomology.

Explicit calculations of sl(N) homology match the cohomology of SU(N) representation spaces.

The approach simplifies understanding of the homological invariants of T(2,m) knots.

Abstract

The sl(N) homology of the torus knot or link T(2,m) may be calculated explicitly. By direct comparison, the result is isomorphic to the cohomology of a naturally associated space of SU(N) representations of the knot group. In honor of Tom Mrowka's 60th birthday, we explain how the Gysin exact sequence may be used to show that these groups are isomorphic without explicitly calculating them.