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Arc presentations of knots and links. (English) Zbl 0904.57004

Jones, Vaughan F. R. (ed.) et al., Knot theory. Proceedings of the mini-semester, Warsaw, Poland, July 13–August 17, 1995. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 42, 57-64 (1998).
Imagine a finite number of half-planes in \(\mathbb{R}^3\) which meet – like an open book – in a single line, called the axis or the binding. Then imagine a knot or link embedded into these planes such that it meets each half-plane in a single arc. Such an embedding is called an arc presentation of the link \(L\). The minimum number of planes required for such a presentation clearly is a link invariant, called the arc index of \(L\), \(\alpha (L)\).
According to the author, arc presentations have been used as a tool since the very early beginnings of knot theory, but no one seems to have investigated them closer until recently; the article gives an overview over first results in this area which have been produced by Ian Nutt and the author himself [P. R. Cromwell, Topology Appl. 64, No. 1, 37-58 (1995; Zbl 0845.57004); P. R. Cromwell and I. J. Nutt, Math. Proc. Camb. Philos. Soc. 119, No. 2, 309-319 (1996; Zbl 0860.57007); I. J. Nutt, Braid index of satellite links, PhD thesis, Univ. Liverpool (1995); Embedding knots and links in an open book III: On the braid index of satellite links, Preprint Univ. Liverpool (1995); J. Knot Theory Ramifications 6, No. 1, 61-77 (1997; Zbl 0879.57009)]. Subjects treated are:
(1) Several methods for description of arc presentations.
(2) Properties of the arc index: behaviour under distant union and connected sum; lower bounds relating the arc index to the crossing number, the braid index and the breadth of the HOMFLY and Kauffman polynomials.
(3) For 2-bridge knots, the result \(\alpha (K)= c(K)+2\) holds where \(\alpha\) denotes the arc index and \(c\) the crossing number.
(4) For alternating knots with at most ten crossings computer calculations show that the same equation holds. Furthermore, for an alternating knot the author can show that \(\alpha (K)\geq c(K)+2\). He conjectures that equality holds for all alternating links.
For the entire collection see [Zbl 0890.00048].

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
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