Some links with non-adequate minimal-crossing diagrams

1992 ◽  
Vol 111 (2) ◽  
pp. 283-289 ◽  
Author(s):  
Masao Hara ◽  
Makoto Yamamoto
Keyword(s):  
Crossing Number ◽  
Link Diagram ◽  
Polynomial Invariants ◽  
Crossing Numbers ◽  
Almost All ◽  
Alternating Link ◽  
Torus Links

To investigate invariants of links derived from their diagrams, the recent new polynomial invariants of links play important roles. Murasugi6, 7, Kauffman3 and Thistlethwaite 9 independently showed that the number of crossings in a proper connected alternating diagram of a link is the minimal-crossing number of the link and that the writhe of the diagram is invariant. Murasugi 8 also determined the minimal-crossing number of torus links. In 5, Lickorish and Thistlethwaite introduced the concept of an adequate link diagram and showed that the number of crossings in an adequate diagram of a semi-alternating link is the minimal-crossing number of the link. They also determined the minimal-crossing number of almost all Montesinos links. In this paper we show that for some links represented by plats and braids which are not adequate, the numbers of crossings in the diagrams are the minimal-crossing numbers of the links.

THE ADDITIVITY OF CROSSING NUMBERS

2004 ◽  
Vol 13 (07) ◽  
pp. 857-866 ◽  
Author(s):  
YUANAN DIAO
Keyword(s):  
Knot Theory ◽  
Difficult Problem ◽  
Crossing Number ◽  
The Other ◽  
Connected Sum ◽  
Torus Knots ◽  
Crossing Numbers ◽  
Weaker Conditions ◽  
Alternating Links ◽  
Alternating Link

It has long been conjectured that the crossing numbers of links are additive under the connected sum of links. This is a difficult problem in knot theory and has been open for more than 100 years. In fact, many questions of much weaker conditions are still open. For instance, it is not known whether Cr(K1#K2)≥Cr(K1) or Cr(K1#K2)≥Cr(K2) holds in general, here K1#K2 is the connected sum of K1 and K2 and Cr(K) stands for the crossing number of the link K. However, for alternating links K1 and K2, Cr(K1#K2)=Cr(K1)+Cr(K2) does hold. On the other hand, if K1 is an alternating link and K2 is any link, then we have Cr(K1#K2)≥Cr(K1). In this paper, we show that there exists a wide class of links over which the crossing number is additive under the connected sum operation. This class is different from the class of all alternating links. It includes all torus knots and many alternating links. Furthermore, if K1 is a connected sum of any given number of links from this class and K2 is a non-trivial knot, we prove that Cr(K1#K2)≥Cr(K1)+3.

A geometric proof that alternating knots are non-trivial

1991 ◽  
Vol 109 (3) ◽  
pp. 425-431 ◽  
Author(s):  
William Menasco ◽  
Morwen Thistlethwaite
Keyword(s):  
Jones Polynomial ◽  
Related Result ◽  
Slight Variation ◽  
Geometric Proof ◽  
Link Diagram ◽  
Combinatorial Argument ◽  
Algebraic Invariant ◽  
Almost All ◽  
Alternating Links ◽  
Alternating Link

There are many proofs in the literature of the non-triviality of alternating, classical links in the 3-sphere, but almost all use a combinatorial argument involving some algebraic invariant, namely the determinant [1], the Alexander polynomial [3], the Jones polynomial [5], and, in [6], the Q-polynomial of Brandt–Lickorish–Millett. Indeed, alternating links behave remarkably well with respect to these and other invariants, but this fact has not led to any significant geometric understanding of alternating link types. Therefore it is natural to seek purely geometric proofs of geometric properties of these links. Gabai has given in [4] a striking geometric proof of a related result, also proved earlier by algebraic means in [3], namely that the Seifert surface obtained from a reduced alternating link diagram by Seifert's algorithm has minimal genus for that link. Here, we give an elementary geometric proof of non-triviality of alternating knots, using a slight variation of the techniques set forth in [7, 8]. Note that if L is a link of more than one component and some component of L is spanned by a disk whose interior lies in the complement of L, then L is a split link, i.e. it is separated by a 2-sphere in S3\L; thus we do not consider alternating links of more than one component here, as it is proved in [7] that a connected alternating diagram cannot represent a split link.

THE POLE DIAGRAM AND THE MIYAZAWA POLYNOMIAL

2008 ◽  
Vol 19 (02) ◽  
pp. 193-207 ◽  
Author(s):  
ATSUSHI ISHII
Keyword(s):  
Crossing Number ◽  
Virtual Link ◽  
Pole Diagram ◽  
Crossing Numbers ◽  
Link Diagrams ◽  
Bracket Polynomial ◽  
Alternating Link ◽  
Simple Definition

We introduce the pole diagram, which helps to retrieve information from a knot diagram when we smooth crossings. By using the notion, we define a bracket polynomial for the Miyazawa polynomial. The bracket polynomial gives a simple definition and evaluation for the Miyazawa polynomial. Then we show that the virtual crossing number of a virtualized alternating link is determined by its diagram. Furthermore, we construct infinitely many virtual link diagrams which attain the minimal real and virtual crossing numbers together.

Cyclic permutations and crossing numbers of join products of two symmetric graphs of order six

2019 ◽  
Vol 35 (2) ◽  
pp. 137-146
Author(s):  
STEFAN BEREZNY ◽  
MICHAL STAS ◽  
◽  
Keyword(s):  
Crossing Number ◽  
Crossing Numbers ◽  
Symmetric Graphs ◽  
Isolated Vertex ◽  
Join Of Graphs ◽  
Discrete Graph

The main purpose of this article is broaden known results concerning crossing numbers for join of graphs of order six. We give the crossing number of the join product G + Dn, where the graph G consists of one 5-cycle and of one isolated vertex, and Dn consists on n isolated vertices. The proof is done with the help of software that generates all cyclic permutations for a given number k, and creates a new graph COG for calculating the distances between all vertices of the graph. Finally, by adding some edges to the graph G, we are able to obtain the crossing numbers of the join product with the discrete graph Dn and with the path Pn on n vertices for other two graphs.

Entanglement complexity of graphs in Z3

1992 ◽  
Vol 111 (1) ◽  
pp. 75-91 ◽  
Author(s):  
C. E. Soteros ◽  
D. W. Sumners ◽  
S. G. Whittington
Keyword(s):  
Crossing Number ◽  
Equivalence Classes ◽  
Closed Curve ◽  
Good Measure ◽  
Average Value ◽  
Cut Edge ◽  
Expected Complexity ◽  
Almost All

AbstractIn this paper we are concerned with questions about the knottedness of a closed curve of given length embedded in Z3. What is the probability that such a randomly chosen embedding is knotted? What is the probability that the embedding contains a particular knot? What is the expected complexity of the knot? To what extent can these questions also be answered for a graph of a given homeomorphism type?We use a pattern theorem due to Kesten 12 to prove that almost all embeddings in Z3 of a sufficiently long closed curve contain any given knot. We introduce the idea of a good measure of knot complexity. This is a function F which maps the set of equivalence classes of embeddings into 0, ). The F measure of the unknot is zero, and, generally speaking, the more complex the prime knot decomposition of a given knot type, the greater its F measure. We prove that the average value of F diverges to infinity as the length (n) of the embedding goes to infinity, at least linearly in n. One example of a good measure of knot complexity is crossing number.Finally we consider similar questions for embeddings of graphs. We show that for a fixed homeomorphism type, as the number of edges n goes to infinity, almost all embeddings are knotted if the homeomorphism type does not contain a cut edge. We prove a weaker result in the case that the homeomorphism type contains at least one cut edge and at least one cycle.

scholarly journals Minimal number of crossings in strong product of paths

2013 ◽  
Vol 29 (1) ◽  
pp. 27-32
Author(s):  
MARIAN KLESC ◽  
◽  
JANA PETRILLOVA ◽  
MATUS VALO ◽  
◽  
...  
Keyword(s):  
Crossing Number ◽  
Cartesian Products ◽  
Crossing Numbers ◽  
Strong Product ◽  
Product Of Graphs

The crossing number cr(G) of a graph G is the minimal number of crossings over all drawings of G in the plane. The exact crossing number is known only for few specific families of graphs. Cartesian products of two graphs belong to the first families of graphs for which the crossing number has been studied. Some results concerning crossing numbers are also known for join products of two graphs. In the paper, we start to collect the crossing numbers for the strong product of graphs, namely for the strong product of two paths.

scholarly journals Braid index bounds ropelength from below

2020 ◽  
Vol 29 (04) ◽  
pp. 2050019
Author(s):  
Yuanan Diao
Keyword(s):  
Lower Bound ◽  
General Formula ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Braid Index ◽  
Alternating Links

For an unoriented link [Formula: see text], let [Formula: see text] be the ropelength of [Formula: see text]. It is known that in general [Formula: see text] is at least of the order [Formula: see text], and at most of the order [Formula: see text] where [Formula: see text] is the minimum crossing number of [Formula: see text]. Furthermore, it is known that there exist families of (infinitely many) links with the property [Formula: see text]. A long standing open conjecture states that if [Formula: see text] is alternating, then [Formula: see text] is at least of the order [Formula: see text]. In this paper, we show that the braid index of a link also gives a lower bound of its ropelength. More specifically, we show that there exists a constant [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] is the largest braid index among all braid indices corresponding to all possible orientation assignments of the components of [Formula: see text] (called the maximum braid index of [Formula: see text]). Consequently, [Formula: see text] for any link [Formula: see text] whose maximum braid index is proportional to its crossing number. In the case of alternating links, the maximum braid indices for many of them are proportional to their crossing numbers hence the above conjecture holds for these alternating links.

THE HOMFLY POLYNOMIAL FOR TORUS LINKS FROM CHERN–SIMONS GAUGE THEORY

1995 ◽  
Vol 10 (07) ◽  
pp. 1045-1089 ◽  
Author(s):  
J. M. F. LABASTIDA ◽  
M. MARIÑO
Keyword(s):  
Gauge Theory ◽  
Gauge Group ◽  
Homfly Polynomial ◽  
Fundamental Representation ◽  
Torus Knots ◽  
Polynomial Invariants ◽  
Chern Simons ◽  
Knots And Links ◽  
Torus Links

Polynomial invariants corresponding to the fundamental representation of the gauge group SU(N) are computed for arbitrary torus knots and links in the framework of Chern–Simons gauge theory making use of knot operators. As a result, a formula for the HOMFLY polynomial for arbitrary torus links is presented.

scholarly journals Crossing Numbers and Hard Erdős Problems in Discrete Geometry

1997 ◽  
Vol 6 (3) ◽  
pp. 353-358 ◽  
Author(s):  
LÁSZLÓ A. SZÉKELY
Keyword(s):  
Lower Bound ◽  
Discrete Geometry ◽  
Crossing Number ◽  
Plane Geometry ◽  
Crossing Numbers ◽  
Minimum Number

We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.

scholarly journals Recurrent Generalization of F-Polynomials for Virtual Knots and Links

Symmetry ◽  
2021 ◽  
Vol 14 (1) ◽  
pp. 15
Author(s):  
Amrendra Gill ◽  
Maxim Ivanov ◽  
Madeti Prabhakar ◽  
Andrei Vesnin
Keyword(s):  
Local Symmetry ◽  
Weight Functions ◽  
Virtual Link ◽  
Link Diagram ◽  
Polynomial Invariants ◽  
Knot Invariants ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
Knots And Links

F-polynomials for virtual knots were defined by Kaur, Prabhakar and Vesnin in 2018 using flat virtual knot invariants. These polynomials naturally generalize Kauffman’s affine index polynomial and use smoothing in the classical crossing of a virtual knot diagram. In this paper, we introduce weight functions for ordered orientable virtual and flat virtual links. A flat virtual link is an equivalence class of virtual links with respect to a local symmetry changing a type of classical crossing in a diagram. By considering three types of smoothing in classical crossings of a virtual link diagram and suitable weight functions, there is provided a recurrent construction for new invariants. It is demonstrated by explicit examples that newly defined polynomial invariants are stronger than F-polynomials.

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