Exact triangles for SO(3) instanton homology of webs

TL;DR

This paper establishes a skein exact triangle for SO(3) instanton homology of webs and connects it to the conjecture relating the homology rank to Tait colorings, advancing understanding of web invariants.

Contribution

The authors prove a skein exact triangle and realize the octahedral axiom for SO(3) instanton homology of webs, providing new tools for studying web invariants.

Findings

Established a skein exact triangle for the instanton homology.

Realized the octahedral axiom within the framework.

Connected the homology rank to Tait colorings for planar webs.

Abstract

The SO(3) instanton homology recently introduced by the authors associates a finite-dimensional vector space over the field of two elements to every embedded trivalent graph (or "web"). The present paper establishes a skein exact triangle for this instanton homology, as well as a realization of the octahedral axiom. From the octahedral diagram, one can derive equivalent reformulations of the authors' conjecture that, for planar webs, the rank of the instanton homology is equal to the number of Tait colorings.