Kontsevich integral for knots and Vassiliev invariants
Petr Dunin-Barkowski, Alexey Sleptsov, Andrey Smirnov
TL;DR
This paper reviews the quantum field theory approach to knot theory, focusing on the Kontsevich integral and Vassiliev invariants, providing computational methods and experimental insights.
Contribution
It introduces a combinatorial method for calculating Vassiliev invariants from the Kontsevich integral suitable for computer implementation.
Findings
Explicit computational examples of Vassiliev invariants
Representation of invariants using arrow diagrams
Discussion of gauge choices affecting calculations
Abstract
We review quantum field theory approach to the knot theory. Using holomorphic gauge we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way which can be programmed on a computer. We discuss experimental results and temporal gauge considerations which lead to representation of Vassiliev invariants in terms of arrow diagrams. Explicit examples and computational results are presented.
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