In this talk, some basic concepts in knot theory such as the
classification/tabulation of knots, knot projections, minimum crossing
numbers of knots will first be introduced with brief discussions,
followed by the introduction of the concepts/definitions of the
ropelength of a knot and invariants of knots. Results concerning the
ropelength of knots will be reviewed briefly. For example, for any
given knot \(K\), it is known that its ropelength \(L(K)\) is at least of
the order \(O((Cr(K))^{3/4})\), and at most of the order
\(O(Cr(K)\ln^{5}(Cr(K))\) where \(Cr(K)\) is the minimum crossing number
of \(K\), while it is known that there exist families of (infinitely
many) links with the property \(L(K)=O(Cr(K))\). A long standing open
conjecture states that if \(K\) is alternating, then \(L(K)\) is at least
of the order \(O(Cr(K))\). The rest of the talk will be devoted to a
recent result which shows that the braid index of a knot (a well-known
and well-studied knot invariant), also provides a lower bound for the
ropelength of the knot. In the case of alternating knots, the braid
indices for many of them are proportional to their crossing numbers
hence the above conjecture holds for these alternating knots.
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