Colloquium

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.