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arXiv · 2608.06372

Squarefree Matrix Formulas for the CWR Invariant of Alternating Knots and Links

Michal Jablonowski
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Abstract

We give weighted-matrix formulas for the components of the $CWR$ invariant of oriented non-split alternating links. After recalling the known trace formulas for $CWR_{2}$ and $CWR_{3}$, we give a construction uniform in $k$: attaching an independent commuting variable to each vertex of a consolidated Tait graph and extracting the squarefree part of the resulting trace isolates simple cycles from closed walks. This yields a formula for $CWR_k$ for every $k\ge 3$, a log-determinant generating polynomial for each of the two Tait graphs, and an equivalent Moebius-inversion formula over principal submatrices. Specializing the uniform formula, we obtain explicit closed weighted formulas for $CWR_{4}$ and $CWR_{5}$. We also record a bipartiteness criterion for the vanishing of all odd components and a characteristic-polynomial formula for the unweighted specialization of the first nonvanishing odd component. The graph-theoretic constructions apply to arbitrary finite simple loopless weighted graphs; the alternating-link hypothesis enters through the invariance theorem for $CWR$.

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Michal Jablonowski. 2026-08-06. Squarefree Matrix Formulas for the CWR Invariant of Alternating Knots and Links. https://arxiv.org/abs/2608.06372

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