SU(3) instanton homology for webs and foams

TL;DR

This paper develops an SU(3) instanton homology theory for webs and foams, establishing skein relations and connecting the homology's dimension to Tait colorings in planar cases, extending previous SO(3) work.

Contribution

It introduces an SU(3) gauge-theoretic homology for webs and foams, with new skein exact triangles and a novel connection to Tait colorings and the Yamada polynomial.

Findings

SU(3) homology counts Tait colorings for planar webs

Skein exact triangles are established for the homology

Euler characteristic relates to signed Tait colorings and Yamada polynomial

Abstract

An instanton homology is constructed for webs and foams, using gauge theory with structure group SU(3), adapting previous work of the authors for the SO(3) case. Skein exact triangles are established, and using an eigenspace decomposition arising from operators associated to the edges, it is shown that the dimension of the SU(3) homology counts Tait colorings when the web is planar. Unlike the SO(3) case, the SU(3) homology is mod-2 graded. Its Euler characteristic can be interpreted as a signed count of Tait colorings, or equivalently as the value at 1 of the Yamada polynomial invariant. Some examples and variants of the construction are also discussed.