Combinatorial Formulae for Vassiliev Invariants from Chern-Simons Gauge Theory

TL;DR

This paper explores the perturbative expansion of Wilson loops in Chern-Simons theory, introducing kernels of Vassiliev invariants, and provides new combinatorial formulas for invariants up to order four.

Contribution

It introduces the concept of kernels for Vassiliev invariants, proposes their reconstruction, and derives new combinatorial expressions for invariants at orders three and four.

Findings

Kernels depend on knot projections but differ from invariants by vanishing terms.

Reconstruction of Vassiliev invariants from kernels is conjectured.

New combinatorial formulas for invariants at order four are presented.

Abstract

We analyse the perturbative series expansion of the vacuum expectation value of a Wilson loop in Chern-Simons gauge theory in the temporal gauge. From the analysis emerges the notion of the kernel of a Vassiliev invariant. The kernel of a Vassiliev invariant of order n is not a knot invariant, since it depends on the regular knot projection chosen, but it differs from a Vassiliev invariant by terms that vanish on knots with n singular crossings. We conjecture that Vassiliev invariants can be reconstructed from their kernels. We present the general form of the kernel of a Vassiliev invariant and we describe the reconstruction of the full primitive Vassiliev invariants at orders two, three and four. At orders two and three we recover known combinatorial expressions for these invariants. At order four we present new combinatorial expressions for the two primitive Vassiliev invariants present at this order.