Abstract:We add twists to a band of a genus-one Seifert surface, producing a knot $K(\ell,0)$. We prove $K$ and $K(\ell,0)$ have $S$-equivalent Seifert matrices if and only if the $(2,2)$-entry of the Seifert matrix vanishes and the sum of off-diagonal entries divides $\ell$. The necessity follows from the Alexander polynomial and a norm argument proving triviality of the $S$-equivalence subgroup $\mathcal{S}^+$ in the class group of binary quadratic forms (Aka--Feller--Miller--Wieser); sufficiency is an explicit $\Lambda_1$-operation. The Jones polynomial distinguishes the knots when $V(K)\neq1$, yielding infinite families of $S$-equivalent but inequivalent genus-one knots, illustrated by $9_{46}$. Also in this paper, we provide a partial answer for Problem~1.6 in Kirby's problem list (K3) and Problem~7.7 of Aka--Feller--Miller--Wieser.
From: Jun Wang [view email]
[v1]
Mon, 25 May 2026 00:13:18 UTC (19 KB)
[v2]
Fri, 3 Jul 2026 07:19:02 UTC (28 KB)
[v3]
Wed, 26 Aug 2026 08:26:55 UTC (27 KB)
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