Abstract:The partial-dual Euler-genus polynomial was defined by Gross, Mansour, and Tucker to analyze how the Euler genus of a ribbon graph changes under partial duality, a generalization of Euler-Poincaré duality introduced by Chmutov. The twist polynomial defined by Yan and Jin extends the partial-dual Euler-genus polynomial to a polynomial on delta-matroids. We derive a recursion formula for the twist polynomial of a one-point join of looped simple graphs -- equivalently, normal, binary delta-matroids. Our recursion applies to the partial-dual Euler-genus polynomial as a special case, where it generalizes a recursion obtained by Yan and Jin. We obtain relations for the twist polynomial on looped simple graphs evaluated at $-1/2$ and for the twist polynomial of a graph with a single looped vertex. A characterization is given for the feasible sets of the delta-matroid associated to a one-point join of looped simple graphs. We show that Yan and Jin's recursion extends to the twist polynomial on delta-matroids.
From: Charlton Li [view email]
[v1]
Wed, 12 Nov 2025 17:34:35 UTC (19 KB)
[v2]
Wed, 5 Aug 2026 22:26:39 UTC (20 KB)
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