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.
Scheduled for: 2025-08-13 08:30 AM: Graphs Session Talk #4.1 in HUMB 146
Status: Accepted
Collection: Topological Graph Theory
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