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SUMMARY:Yuanan Diao (UNC Charlotte)
DTSTART:20210405T160000Z
DTEND:20210405T170000Z
DTSTAMP:20260423T005835Z
UID:TG_ET/16
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/TG_ET/16/">T
 he ropelength of knots</a>\nby Yuanan Diao (UNC Charlotte) as part of Topo
 logy and geometry: extremal and typical\n\n\nAbstract\nThe ropelength $R(K
 )$ of a knot $K$ is the minimum length of a unit thickness rope needed to 
 tie the knot. If $K$ is alternating\, it is conjectured that $R(K)\\ge a {
 \\rm{Cr}}(K)$ for some constant $a>0$\, where ${\\rm{Cr}}(K)$ is the minim
 um crossing number of $K$. In this talk I will first give a brief introduc
 tion to the ropelength problem. I will then show that there exists a const
 ant $a_0>0$ such that $R(K)\\ge a_0 \\textbf{b}(K)$ for any knot $K$\, whe
 re $\\textbf{b}(K)$ is the braid index of $K$. It follows that if $\\textb
 f{b}(K)\\ge a_1 {\\rm{Cr}}(K)$ for some constant $a_1>0$\, then $R(K)\\ge 
 a_0 a_1 {\\rm{Cr}}(K)=a {\\rm{Cr}}(K)$. However if $\\textbf{b}(K)$ is sma
 ll compared to ${\\rm{Cr}}(K)$ (in fact there are alternating knots with a
 rbitrarily large crossing numbers but fixed braid indices)\, then this res
 ult cannot be applied directly. I will show that this result can in fact b
 e applied in an indirect way to prove that the conjecture holds for a larg
 e class of alternating knots\, regardless what their braid indices are.\n
LOCATION:https://researchseminars.org/talk/TG_ET/16/
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