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Essential properties of the Tutte polynomial
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
V. Jones [668] started a revolution in knot theory when he introduced the Jones polynomial, a link invariant defined via von Neumann algebras that is extremely powerful—it distinguishes many pairs of knots/links. However, the real breakthrough came only a little later, when Kauffman [681] found a combinatorial way of using link diagrams to calculate (and re-prove the existence of) the Jones polynomial, based on an expansion (or, equivalently, a recurrence) involving “resolving” a crossing in two ways, as in Figure 3.3. Thistlethwaite [1061] noticed that this Kauffman bracket is extremely closely related to the Tutte polynomial. Indeed, for alternating links, where the crossings alternate over, under, over, under, etc. as one moves along a strand of the link, the Kauffman bracket, and hence the Jones polynomial, is given by an evaluation of the Tutte polynomial.
The knowledge of knots: an interdisciplinary literature review
Published in Spatial Cognition & Computation, 2019
Paulo E. Santos, Pedro Cabalar, Roberto Casati
Topological Knot Theory (or simply Knot Theory) is a well-established discipline in mathematics (Menasco, 2005) that has a long track of theoretical results derived from the fields of combinatorics, topology and group theory, and has provided important contributions to theoretical physics, biology and chemistry (Kauffman, 2005; Murasugi, 1996a). Topological problems considering strings and knots have been studied since Gauss in the early nineteenth century (van de Griend, 1998). Nowadays, Knot Theory constitutes a very prolific area in topology, currently represented by one specialized journal (the Journal of Knot Theory and its Ramifications, since in 1992) and several textbooks (Adams, 1994; Crowell & Fox, 2008; Kauffman, 2006; Lackenby, 2016)3A quick search on MIT library brought over one million references of books and half-a-million of articles related to Knot Theory. making a complete up-to-date review of this area virtually impossible. Still, in the interest of contextualizing this research, the present section relies on existing reviews (Lackenby, 2016; van de Griend, 1998). Further literature surveys are given in (Kawauchi, 1996; Menasco & Thistlethwaite, 2005) and the excellent historical overview in (van de Griend, 1998).