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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
In topology, a knot is a simple closed curve in ℝ3, and a link is a finite set of pairwise disjoint simple closed curves in ℝ3. Two knots or links are equivalent if one can be deformed into the other in the natural way. The unknot is the “unknotted” knot, for example, a geometric circle. Often, it is easy to see that two links are equivalent—simply exhibit a deformation. But how can one tell that two links are not equivalent, or that a given knot is not the unknot? Sometimes there is an easy answer based on a simple invariant, such as the number of components in a link, or the “linking number” (closely related to winding number). But in general this is a very hard problem.
The knowledge of knots: an interdisciplinary literature review
Published in Spatial Cognition & Computation, 2019
Paulo E. Santos, Pedro Cabalar, Roberto Casati
The scientific literature on knots usually distinguishes three basic types of string entanglements: hitches, braids, and knots. Usually, they are (informally) defined as follows: hitches are a special kind of knots used to fasten a rope around another object (usually a post or another rope); braids are entanglements of a number of strings generated by twisting motions, so that the direction of each string remains the same. The general term for the knot is used to represent entanglements of strings capable of holding their own shape, regardless of their relation with external objects. We also find the term link to represent an intertwined, but non-intersecting, collection of knots. Distinct basic definitions of these concepts are assumed by the distinct disciplines that investigate knots2There are further distinctions commonly used by knot practitioners (such as slip knots, or nooses–that, topologically, are simple loops in the rope’s end) but adding a description of every one of them is a subject orthogonal to the interests of this paper.