Ineffective sets and the region crossing change operation

2020 ◽  
Vol 29 (03) ◽  
pp. 2050010
Author(s):  
Miles Clikeman ◽  
Rachel Morris ◽  
Heather M. Russell
Keyword(s):  
Upper Bounds ◽  
The Other ◽  
Link Diagram ◽  
Complete Collection ◽  
Link Diagrams ◽  
Change Operation ◽  
Knot Diagrams

Region crossing change (RCC) is an operation on link diagrams in which all crossings incident to a selected region are changed. Two diagrams are called RCC-equivalent if one can be transformed to the other via a sequence of RCCs. RCC is an unknotting operation but not an unlinking operation. A set of regions of a diagram is called ineffective if RCCs at every region in that set have no net effect on the crossings of the diagram. The main result of this paper is a construction of the complete collection of ineffective sets for any link diagram. This involves a combination of linear algebraic and diagrammatic techniques including a generalization of checkerboard shading called tricoloring. Using this construction of ineffective sets, we provide sharp upper bounds on the maximum number of RCCs needed to transform between RCC-equivalent knot diagrams and reduced 2- and 3-component link diagrams with fixed underlying projections.

REIDEMEISTER MOVES FOR SURFACE ISOTOPIES AND THEIR INTERPRETATION AS MOVES TO MOVIES

1993 ◽  
Vol 02 (03) ◽  
pp. 251-284 ◽  
Author(s):  
J. SCOTT CARTER ◽  
MASAHICO SAITO
Keyword(s):  
Cross Sections ◽  
Height Function ◽  
The Other ◽  
Projection Direction ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Fixed Direction ◽  
The Cross ◽  
Knot Diagrams

A movie description of a surface embedded in 4-space is a sequence of knot and link diagrams obtained from a projection of the surface to 3-space by taking 2-dimensional cross sections perpendicular to a fixed direction. In the cross sections, an immersed collection of curves appears, and these are lifted to knot diagrams by using the projection direction from 4-space. We give a set of 15 moves to movies (called movie moves) such that two movies represent isotopic surfaces if and only if there is a sequence of moves from this set that takes one to the other. This result generalizes the Roseman moves which are moves on projections where a height function has not been specified. The first 7 of the movie moves are height function parametrized versions of those given by Roseman. The remaining 8 are moves in which the topology of the projection remains unchanged.

Linking invariants of even virtual links

2017 ◽  
Vol 26 (12) ◽  
pp. 1750072 ◽  
Author(s):  
Haruko A. Miyazawa ◽  
Kodai Wada ◽  
Akira Yasuhara
Keyword(s):  
Lower Bound ◽  
The Other ◽  
Virtual Link ◽  
Link Diagram ◽  
Linking Number ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Virtual Links ◽  
The Difference

A virtual link diagram is even if the virtual crossings divide each component into an even number of arcs. The set of even virtual link diagrams is closed under classical and virtual Reidemeister moves, and it contains the set of classical link diagrams. For an even virtual link diagram, we define a certain linking invariant which is similar to the linking number. In contrast to the usual linking number, our linking invariant is not preserved under the forbidden moves. In particular, for two fused isotopic even virtual link diagrams, the difference between the linking invariants of them gives a lower bound of the minimal number of forbidden moves needed to deform one into the other. Moreover, we give an example which shows that the lower bound is best possible.

Splitting number of augmented links

2018 ◽  
Vol 27 (06) ◽  
pp. 1850038
Author(s):  
Darlan Girao
Keyword(s):  
Link Diagram ◽  
Link Diagrams ◽  
Linking Numbers ◽  
Splitting Number ◽  
Knot Diagrams

We completely determine the splitting number of augmented links arising from knot and link diagrams in which each twist region has an even number of crossings. In the case of augmented links obtained from knot diagrams, we show that the splitting number is given by the size of a maximal collection of Boromean sublinks, any two of which have one component in common. The general case is stablished by considering the linking numbers between components of the augmented links. We also discuss the case when the augmented link arises from a link diagram in which twist regions may have an odd number of crossings.

An upper bound for the breadth of the Jones polynomial

1988 ◽  
Vol 103 (3) ◽  
pp. 451-456 ◽  
Author(s):  
Morwen B. Thistlethwaite
Keyword(s):  
Simple Proof ◽  
Upper Bound ◽  
Recent Article ◽  
Jones Polynomial ◽  
Upper Bounds ◽  
The Other ◽  
Link Diagram ◽  
Kauffman Bracket ◽  
Bracket Polynomial ◽  
Connected Diagram

In the recent article [2], a kind of connected link diagram called adequate was investigated, and it was shown that the Jones polynomial is never trivial for such a diagram. Here, on the other hand, upper bounds are considered for the breadth of the Jones polynomial of an arbitrary connected diagram, thus extending some of the results of [1,4,5]. Also, in Theorem 2 below, a characterization is given of those connected, prime diagrams for which the breadth of the Jones polynomial is one less than the number of crossings; recall from [1,4,5] that the breadth equals the number of crossings if and only if that diagram is reduced alternating. The article is concluded with a simple proof, using the Jones polynomial, of W. Menasco's theorem [3] that a connected, alternating diagram cannot represent a split link. We shall work with the Kauffman bracket polynomial 〈D〉 ∈ Z[A, A−1 of a link diagram D.

scholarly journals Up–down colorings of virtual-link diagrams and the necessity of Reidemeister moves of type II

2017 ◽  
Vol 26 (12) ◽  
pp. 1750073 ◽  
Author(s):  
Kanako Oshiro ◽  
Ayaka Shimizu ◽  
Yoshiro Yaguchi
Keyword(s):  
Type Ii ◽  
Virtual Link ◽  
Link Diagram ◽  
Virtual Knot ◽  
Reidemeister Moves ◽  
Link Diagrams ◽  
Classical Link ◽  
Minimum Number ◽  
Knot Diagrams ◽  
Text Component

We introduce an up–down coloring of a virtual-link (or classical-link) diagram. The colorabilities give a lower bound of the minimum number of Reidemeister moves of type II which are needed between two [Formula: see text]-component virtual-link (or classical-link) diagrams. By using the notion of a quandle cocycle invariant, we give a method to detect the necessity of Reidemeister moves of type II between two given virtual-knot (or classical-knot) diagrams. As an application, we show that for any virtual-knot diagram [Formula: see text], there exists a diagram [Formula: see text] representing the same virtual-knot such that any sequence of generalized Reidemeister moves between them includes at least one Reidemeister move of type II.

ON TANGLES AND MATROIDS

2005 ◽  
Vol 14 (07) ◽  
pp. 919-929 ◽  
Author(s):  
STEPHEN HUGGETT
Keyword(s):  
Tensor Product ◽  
Planar Graph ◽  
Planar Graphs ◽  
Knot Theory ◽  
Tutte Polynomial ◽  
The Other ◽  
Link Diagram ◽  
Polynomial Invariant ◽  
Link Diagrams ◽  
Alternating Link

Given matroids M and N there are two operations M ⊕2 N and M ⊗ N. When M and N are the cycle matroids of planar graphs these operations have interesting interpretations on the corresponding link diagrams. In fact, given a planar graph there are two well-established methods of generating an alternating link diagram, and in each case the Tutte polynomial of the graph is related to a polynomial invariant (Jones or Homfly) of the link. Switching from one of these methods to the other corresponds in knot theory to tangle insertion in the link diagrams, and in combinatorics to the tensor product of the cycle matroids of the graphs.

Spanning Properties of Yao and 𝜃-Graphs in the Presence of Constraints

2019 ◽  
Vol 29 (02) ◽  
pp. 95-120 ◽  
Author(s):  
Prosenjit Bose ◽  
André van Renssen
Keyword(s):  
Line Segment ◽  
Large Family ◽  
Upper Bounds ◽  
The Other ◽  
Additional Property ◽  
Graph Partitions ◽  
Set Of Points ◽  
The Given

We present improved upper bounds on the spanning ratio of constrained [Formula: see text]-graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text], and adds an edge to the closest vertex in each cone. Constrained Yao-graphs have the additional property that no edge properly intersects any of the given line segment constraints. Constrained [Formula: see text]-graphs are similar to constrained Yao-graphs, but use a different method to determine the closest vertex. We present tight bounds on the spanning ratio of a large family of constrained [Formula: see text]-graphs. We show that constrained [Formula: see text]-graphs with [Formula: see text] ([Formula: see text] and integer) cones have a tight spanning ratio of [Formula: see text], where [Formula: see text] is [Formula: see text]. We also present improved upper bounds on the spanning ratio of the other families of constrained [Formula: see text]-graphs. These bounds match the current upper bounds in the unconstrained setting. We also show that constrained Yao-graphs with an even number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text] and constrained Yao-graphs with an odd number of cones ([Formula: see text]) have spanning ratio at most [Formula: see text]. As is the case with constrained [Formula: see text]-graphs, these bounds match the current upper bounds in the unconstrained setting, which implies that like in the unconstrained setting using more cones can make the spanning ratio worse.

scholarly journals Cyclic coverings of virtual link diagrams

2019 ◽  
Vol 30 (14) ◽  
pp. 1950072 ◽  
Author(s):  
Naoko Kamada
Keyword(s):  
Virtual Link ◽  
Link Diagram ◽  
Cyclic Covering ◽  
Link Diagrams ◽  
Virtual Links ◽  
Cyclic Coverings

A virtual link diagram is called mod [Formula: see text] almost classical if it admits an Alexander numbering valued in integers modulo [Formula: see text], and a virtual link is called mod [Formula: see text] almost classical if it has a mod [Formula: see text] almost classical diagram as a representative. In this paper, we introduce a method of constructing a mod [Formula: see text] almost classical virtual link diagram from a given virtual link diagram, which we call an [Formula: see text]-fold cyclic covering diagram. The main result is that [Formula: see text]-fold cyclic covering diagrams obtained from two equivalent virtual link diagrams are equivalent. Thus, we have a well-defined map from the set of virtual links to the set of mod [Formula: see text] almost classical virtual links. Some applications are also given.

DETERMINING THE COMPONENT NUMBER OF LINKS CORRESPONDING TO LATTICES

2009 ◽  
Vol 18 (12) ◽  
pp. 1711-1726 ◽  
Author(s):  
XIAN'AN JIN ◽  
FENGMING DONG ◽  
ENG GUAN TAY
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Plane Graph ◽  
Link Diagram ◽  
Plane Graphs ◽  
Link Diagrams ◽  
One To One

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.

scholarly journals Regarding Two Conjectures on Clique and Biclique Partitions

10.37236/9564 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Dhruv Rohatgi ◽  
John C. Urschel ◽  
Jake Wellens
Keyword(s):  
Upper Bounds ◽  
The Other ◽  
Lower And Upper Bounds ◽  
Minimum Number ◽  
Other Regarding

For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures – one regarding the maximum possible value of $cp(G) + cp(\overline{G})$ (due to de Caen, Erdős, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $\max_G cp(G) + cp(\overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.

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