S-Equivalence of Band-Twisted Genus One Knots

Summary · qwen2.5:32b

The article proves that a genus-one knot $K$ and its band-twisted variant $K(\ell,0)$ are $S$-equivalent if and only if the $(2,2)$-entry of their Seifert matrix is zero and the sum of off-diagonal entries divides $\ell$. This equivalence is shown to yield infinite families of $S$-equivalent but inequivalent genus-one knots, such as the knot $9_{46}$.

Suggested post angle

This article discusses S-Equivalence of Band-Twisted Genus One Knots, a topic in Mathematics involving algebra and topology.

Excerpt

arXiv:2605.25309v2 Announce Type: replace 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.