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

  • Nancy Scherich
  • Nathaniel Song
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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.

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Cite

BibTeXRIS
BibTeX
@misc{scherich2026combinatorial,
  title = {{Combinatorial Goussarov-Polyak-Viro Formulas for the Linking Number and Low Degree Coefficients of the Conway Polynomial}},
  author = {Nancy Scherich and Nathaniel Song},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20413},
  eprint = {2609.20413},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20413},
}
RIS
TY  - UNPB
TI  - Combinatorial Goussarov-Polyak-Viro Formulas for the Linking Number and Low Degree Coefficients of the Conway Polynomial
AU  - Nancy Scherich
AU  - Nathaniel Song
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20413
UR  - https://arxiv.org/abs/2609.20413
AB  - 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.
ER  -