Combinatorial Goussarov-Polyak-Viro Formulas for the Linking Number and Low Degree Coefficients of the Conway Polynomial

Abstract: The linking number and coefficients of the Conway polynomial are two examples of finite type invariants. Goussarov-Polyak-Viro proved that any finite type knot invariant can be computed in a two step process, where the second step is referred to as the GPV map for the invariant. This paper computes the GPV maps for the linking number and low-degree coefficients of the Conway Polynomial. Chmutov-Khoury-Rossi gave one description of the GPV maps for coefficients of the Conway polynomial in terms of arrow diagrams and state-sum calculations. We take a much more grounded approach using fundamental linear algebra to describe the GPV maps in terms of a basis for the vector space of Gauss diagrams.