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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
The most common way of representing a link is via a link diagram, obtained by projecting the link into the plane in the natural way (avoiding any degenerate points etc.). Rather than give a formal definition, we draw some examples in Figure 3.2. Of course, link diagrams can be viewed as 4-regular plane graphs with some extra information at each “crossing”, namely which strand passes on top. Reidemeister [957], and independently Alexander and Briggs [19], showed that two links are equivalent if and only if their link diagrams can be transformed into each other by a sequence of “Reidemeister moves”, certain simple local operations on the diagram (see Theorem 18.5). Unfortunately, there is no obvious bound on the length of the sequence that may be needed, since Reidemeister moves can increase as well as decrease the complexity of a diagram. Such bounds are a topic of ongoing research; see in particular the paper of Lackenby [739] giving a polynomial bound in the case that the diagram can be reduced to the unknot, as well as a discussion of what is known in other cases.
Proposal of new topology information Face-list for manipulation planning of deformable string tying
Published in SICE Journal of Control, Measurement, and System Integration, 2023
Junxiang Wang, Tomoya Shirakawa, Tomotoshi Watanabe, Yuichiro Toda, Takayuki Matsuno
Reidemeister move and Cross are adopted as a basic unit of shape operations in this study. These are defined in knot theory [17]. There are three types of Reidemeister move, which are named Reidemeister move I, Reidemeister move II and Reidemeister move III. Reidemeister move I is an operation to make a simple loop as shown in Figure 4(a). Reidemeister move II is an operation to intersect a segment with another segment as shown in Figure 4(b). Reidemeister move III is an operation to move a series of segments across a node as shown in Figure 4(c). Cross is an operation to make an endpoint intersect a segment as shown in Figure 4(d). There are two types of Cross, which are Cross of the initial endpoint and Cross of the terminal endpoint. As shown in Figure 4, Reidemeister move and Cross include both operations to add nodes and operations to reduce nodes.
Fertile metastability
Published in Liquid Crystals, 2023
In the first drawing in Figure 11(d), the kink is marked with a red circle and the segments of the loop 2 adjacent to the kink are drawn with lines of different thicknesses because they are not located at the same level: the segment 2l is below the segment 2u. Due to the compression of the wedge, the cholesteric helix is supertwisted and the Peach-Koehler force acting on the dislocation drives the expansion of the loop. The velocity of the expansion is represented by arrows. The second drawing in Figure 11(e) shows that as the kink introduces an additional dissipation, its velocity is lower than that of the loop, , so that it stays more and more behind the loop during its expansion. The kink remains connected to the loop by the elementary Lehmann cluster (a metastable state), i.e. by a pair of associated dislocations, of growing length. The third drawing in Figure 11(f) shows that upon a large enough compression ratio the Lehmann cluster splits into a the new loop labelled 3. N.B: In the theory of knots, this deformation leading to a continuous generation of an additional loop is called the Reidemeister move of type I.
The knowledge of knots: an interdisciplinary literature review
Published in Spatial Cognition & Computation, 2019
Paulo E. Santos, Pedro Cabalar, Roberto Casati
Perhaps the approach to the equivalence problem that permeates the other fields of interest in the present paper (as we shall see below) is the method for deciding the equivalence by means of reducing one-knot representation to another using Reidemeister Moves (Coward & Lackenby, 2011; Reidemeister, 1983). The equivalence problem has a diagrammatic representation, whereby a “knot” is actually assumed to be an equivalence class of knot diagrams. Some operations on the syntactic elements of diagrams allow establishing equivalences in an intuitive and straightforward way. Reidemeister Moves (Reidemeister, 1983) are local modifications in the link diagrams representing knots that preserve the topological properties of the knot. Figure 1 shows the three basic Reidemeister moves, that can be described as follows: move I (Figure 1a) adds or deletes a simple twist in the string; move II (Figure 1b) allows the inclusion (or exclusion) of two crossings in the string; and, move III (Figure 1c) slides a strand of the string from one side of a crossing to the other.