The Ropelengths of Knots
speaker: Yuanan Diao (Department of Mathematics and Statistics, University of North Carolina at Charlotte)abstract: For a knot or link $\mathcal{K}$, let $L(\mathcal{K})$ be the ropelength of $\mathcal{K}$ and $Cr(\mathcal{K})$ be the crossing number of $\mathcal{K}$. In this paper, we show that there exists a constant $a>0$ such that $L(\mathcal{K})\le a Cr(\mathcal{K}) \ln5 (Cr(\mathcal{K}))$ for any $\mathcal{K}$. This result shows that the upper bound of the ropelength of any knot is almost linear in terms of its minimum crossing number, and is a significant improvement over the best known upper bound established previously, which is of the form $L(\mathcal{K})\le O(Cr(\mathcal{K}){\frac{3}{2}})$. The approach taken proves a more general result concerning plane graph embeddings since a knot diagram can be treated as a 4-regular plane graph. More specifically, we prove that any 4-regular plane graph of $n$ vertices can be embedded into the cubic lattice with an embedding length at most of the order $O(n\ln5(n))$, while preserving its topology. The main idea in the proof uses a divide-and-conquer technique to repeatedly subdivide a 4-regular plane graph. The subdivision process and the embedding construction is highly non-trivial because the topology of the knot (or of the graph) must be preserved.
timetable:
Wed 29 Jun, 11:00 - 12:00, Aula Dini
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