Uniqueness of the GPV map for the quadratic Conway coefficient

Determine whether the GPV map \(\omega_{V_2}\) for the quadratic coefficient \(V_2\) of the Conway polynomial is uniquely determined, or whether infinitely many GPV maps arise from the one-parameter family described in Theorem 4.1; equivalently, determine whether the parameter \(t\) has a unique value or admits multiple values.

Background

The paper computes the GPV map ωV2:GD2→Q\omega_{V_2}:GD_2\to\mathbb{Q} associated with the quadratic coefficient V2V_2 of the Conway polynomial on a basis of Gauss diagrams with at most two arrows. The values on one-arrow, nested two-arrow, and disjoint two-arrow diagrams are determined, while the values on the interleaved two-arrow diagrams are described by a one-parameter family involving tt.

The authors’ linear system, constructed from several knots and unknots, leaves one free variable. They therefore do not establish whether the parameter reflects genuine nonuniqueness of the GPV map or merely insufficient equations. Resolving the issue would require either finding additional knot expansions that determine tt uniquely or proving that the stated one-parameter family consists of valid, distinct GPV maps.

References

The results of our computation are inconclusive as to whether there is truly a one-parameter family of GPV maps for the quadratic coefficient of the Conway polynomial, or if we were not able to completely determine the system.

However, we don't expect $\omega_{V_2}$ to be unique! The GPV theorem does not guarantee uniqueness in any sense, and we conjecture that Theorem \ref{thm:main_Conway} accurately describes all possible $\omega_{V_2}$ maps.

For the reminder of this section, we explain some methods we attempted to use to determine a unique value of $t$. We encourage the reader to continue this pursuit to either determine a unique value of $t$, or verify that there are indeed infinitely many possible choices for $t$.

— Combinatorial Goussarov-Polyak-Viro Formulas for the Linking Number and Low Degree Coefficients of the Conway Polynomial  (2609.20413 - Scherich et al., 17 Sep 2026) in Section 4, immediately after the proof of Theorem \ref{thm:main_Conway}, in the discussion beginning “Now, let's address the question: is \(\omega_{V_2}\) unique?”