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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
For a different proof that any reduced (twist-free) alternating knot diagram represents a non-trivial link, based on a connection between the interlace polynomial and the Alexander polynomial (a knot invariant preceding the Jones polynomial), see Balister, Bollobás, Riordan and Scott [68]. A generalization of the Jones polynomial was introduced by Freyd, Yetter, Hoste, Lickorish, Millett and Ocneanu [503], and related independent work was undertaken by Przytycki and Traczyk [939]. A state space expansion of this homflyptpolynomialhomflypt polynomial (or homflypolynomial) was given by Bollobás, Pebody and Weinreich [157]. A state space expansion of this homflyptpolynomialhomflypt polynomial was given by Bollobás, Pebody and Weinreich [157]. Unfortunately, even the homflypt polynomial does not distinguish all knots; two knots related by mutation are known to have the same homflypt polynomial (see [503]). For more on these topics see, for example, the books [2, 772, 815].
The knowledge of knots: an interdisciplinary literature review
Published in Spatial Cognition & Computation, 2019
Paulo E. Santos, Pedro Cabalar, Roberto Casati
The fundamental issue in knot theory is deciding whether two knots or links are equivalent: this is known as the equivalence problem. Most work on this topic is related to the search of knot invariants, which are formal expressions that uniquely represent each knot type, independently of any particular depiction of it.Cha and Livingston (2013) present a table with over 50-knot invariants4A detailed description of knot invariants would justify a survey paper by itself and, therefore, it is outside the scope of this interdisciplinary review. The related Wikipedia entry has an excellent description of some of the most fundamental invariants to date: https://en.wikipedia.org/wiki/Knot_invariant . The importance of knot invariants goes beyond the solution of the equivalence problem, as they represent the fundamental properties of knots.