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Topological Graph Theory

Graphs Session Talk #4.1

Subevent of Graphs Session #4

HUMB 146

Times: 2025 Aug 13 from 08:30AM to 08:55AM (Central Time (US & Canada))

Webinar link: Icon: video Webinar

Flowers of knots

Kouki Taniyama ⟨taniyama@waseda.jp⟩

Abstract:

An $(n,k)$-flower $F(n,k)$ is the shadow of the closure of an $n$-braid $(\sigma_{1}\sigma_{2}\cdots\sigma_{n-1})^{k}$.
C. Lamm and V. O. Manturov independently showed the following: Let $K$ be a knot and $n\geq \mathrm{braid}(K)$. Then $K$ has $F(n,k)$ as its shadow for some $k$. We show the following: Let $K$ be a knot and $k\geq \mathrm{bridge}(K)$. Then $K$ has $F(n,k)$ as its shadow for some $n$. As a corollary, we show $\mathrm{bridge}(K)=\mathrm{lr}(K)$ where $\mathrm{lr}(K)$ is the left-right number of $K$. This gives us a new definition of the bridge number of a knot.

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