scholarly journals Characterization of quasi-alternating Montesinos links

2015 ◽  
Vol 24 (01) ◽  
pp. 1550002 ◽  
Author(s):  
K. Qazaqzeh ◽  
N. Chbili ◽  
B. Qublan
Keyword(s):  
Knot Theory ◽  
Infinite Family ◽  
Rational Tangles ◽  
Alternating Links ◽  
Alternating Link

Let L be a quasi-alternating link at a crossing c. We construct an infinite family of quasi-alternating links from L by replacing the crossing c by a product of rational tangles, each of which extends c. Consequently, we determine an infinite family of quasi-alternating Montesinos links. This family includes all classes of quasi-alternating Montesinos links that have been detected by Widmer [Quasi-alternating Montesinos links, J. Knot Theory Ramifications18(10) (2009) 1459–1469]. We conjecture that this family contains all non-alternating quasi-alternating Montesinos 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.

scholarly journals A GEOMETRIC CHARACTERIZATION OF THE UPPER BOUND FOR THE SPAN OF THE JONES POLYNOMIAL

2011 ◽  
Vol 20 (07) ◽  
pp. 1059-1071
Author(s):  
JUAN GONZÁLEZ-MENESES ◽  
PEDRO M. G. MANCHÓN
Keyword(s):  
Upper Bound ◽  
Knot Theory ◽  
Jones Polynomial ◽  
Geometric Proof ◽  
Link Diagram ◽  
Closed Curves ◽  
Number Of Faces ◽  
Alternating Links ◽  
Connected Diagram

Let D be a link diagram with n crossings, sA and sB be its extreme states and |sAD| (respectively, |sBD|) be the number of simple closed curves that appear when smoothing D according to sA (respectively, sB). We give a general formula for the sum |sAD| + |sBD| for a k-almost alternating diagram D, for any k, characterizing this sum as the number of faces in an appropriate triangulation of an appropriate surface with boundary. When D is dealternator connected, the triangulation is especially simple, yielding |sAD| + |sBD| = n + 2 - 2k. This gives a simple geometric proof of the upper bound of the span of the Jones polynomial for dealternator connected diagrams, a result first obtained by Zhu [On Kauffman brackets, J. Knot Theory Ramifications6(1) (1997) 125–148.]. Another upper bound of the span of the Jones polynomial for dealternator connected and dealternator reduced diagrams, discovered historically first by Adams et al. [Almost alternating links, Topology Appl.46(2) (1992) 151–165.], is obtained as a corollary. As a new application, we prove that the Turaev genus is equal to the number k of dealternator crossings for any dealternator connected diagram.

scholarly journals Generating links that are both quasi-alternating and almost alternating

2020 ◽  
pp. 2050090
Author(s):  
Hamid Abchir ◽  
Mohammed Sabak
Keyword(s):  
Infinite Family ◽  
Crossing Number ◽  
Alternating Links ◽  
Alternating Link

We construct an infinite family of links which are both almost alternating and quasi-alternating from a given either almost alternating diagram representing a quasi-alternating link, or connected and reduced alternating tangle diagram. To do that we use what we call a dealternator extension which consists in replacing the dealternator by a rational tangle extending it. We note that all non-alternating and quasi-alternating Montesinos links can be obtained in that way. We check that all the obtained quasi-alternating links satisfy Conjecture 3.1 of Qazaqzeh et al. (JKTR 22 (6), 2013), that is the crossing number of a quasi-alternating link is less than or equal to its determinant. We also prove that the converse of Theorem 3.3 of Qazaqzeh et al. (JKTR 24 (1), 2015) is false.

scholarly journals Affine Primitive Groups and Semisymmetric Graphs

10.37236/2549 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Hua Han ◽  
Zaiping Lu
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Infinite Family ◽  
Permutation Groups ◽  
Primitive Groups ◽  
The Third ◽  
Semisymmetric Graphs

In this paper, we investigate semisymmetric graphs which arise from affine primitive permutation groups. We give a characterization of such graphs, and then construct an infinite family of semisymmetric graphs which contains the Gray graph as the third smallest member. Then, as a consequence, we obtain a factorization,of the complete bipartite graph $K_{p^{sp^t},p^{sp^t}}$ into connected semisymmetric graphs, where $p$ is an prime, $1\le t\le s$ with $s\ge2$ while $p=2$.

POSITIVE ALTERNATING LINKS ARE POSITIVELY ALTERNATING

2000 ◽  
Vol 09 (01) ◽  
pp. 107-112 ◽  
Author(s):  
TAKUJI NAKAMURA
Keyword(s):  
Alternating Links ◽  
Alternating Link

By using a result of P. R. Cromwell on homogeneous links which contain both positive links and alternating links, we prove that any reduced alternating diagram of a positive alternating link is positive.

scholarly journals GRAPH SMALL CANCELLATION THEORY APPLIED TO ALTERNATING LINK GROUPS

2012 ◽  
Vol 21 (11) ◽  
pp. 1250113
Author(s):  
RÉMI CUNÉO ◽  
HAMISH SHORT
Keyword(s):  
Small Cancellation ◽  
Cancellation Condition ◽  
Small Cancellation Theory ◽  
Link Group ◽  
Finite Sums ◽  
Small Cancellation Condition ◽  
Alternating Links ◽  
Alternating Link

We show that the Wirtinger presentation of a prime alternating link group satisfies a generalized small cancellation condition. This new version of Weinbaum's solution to the word and conjugacy problems for these groups easily extends to finite sums of alternating links.

scholarly journals Writhe-like invariants of alternating links

2021 ◽  
Vol 30 (01) ◽  
pp. 2150004
Author(s):  
Yuanan Diao ◽  
Van Pham
Keyword(s):  
Jones Polynomial ◽  
Link Diagram ◽  
Link Invariant ◽  
Link Invariants ◽  
Link Diagrams ◽  
Alternating Links ◽  
Alternating Link

It is known that the writhe calculated from any reduced alternating link diagram of the same (alternating) link has the same value. That is, it is a link invariant if we restrict ourselves to reduced alternating link diagrams. This is due to the fact that reduced alternating link diagrams of the same link are obtainable from each other via flypes and flypes do not change writhe. In this paper, we introduce several quantities that are derived from Seifert graphs of reduced alternating link diagrams. We prove that they are “writhe-like” invariants, namely they are not general link invariants, but are invariants when restricted to reduced alternating link diagrams. The determination of these invariants are elementary and non-recursive so they are easy to calculate. We demonstrate that many different alternating links can be easily distinguished by these new invariants, even for large, complicated knots for which other invariants such as the Jones polynomial are hard to compute. As an application, we also derive an if and only if condition for a strongly invertible rational link.

A REMARK ON THE DETERMINANT OF QUASI-ALTERNATING LINKS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350031 ◽  
Author(s):  
K. QAZAQZEH ◽  
B. QUBLAN ◽  
A. JARADAT
Keyword(s):  
Crossing Number ◽  
Alternating Links ◽  
Alternating Link

We show that the crossing number of any link that is known to be quasi-alternating is less than or equal to its determinant. Based on this, we conjecture that the crossing number of any quasi-alternating link is less than or equal to its determinant. Thus if this conjecture is true, then it gives a new property of quasi-alternating links and easy obstruction to a link being quasi-alternating.

scholarly journals Khovanov homology for alternating tangles

2014 ◽  
Vol 23 (02) ◽  
pp. 1450013 ◽  
Author(s):  
Dror Bar-Natan ◽  
Hernando Burgos-Soto
Keyword(s):  
Khovanov Homology ◽  
Single Crossing ◽  
Planar Algebra ◽  
Diagonal Condition ◽  
Alternating Links ◽  
Alternating Link

We describe a "concentration on the diagonal" condition on the Khovanov complex of tangles, show that this condition is satisfied by the Khovanov complex of the single crossing tangles [Formula: see text] and [Formula: see text], and prove that it is preserved by alternating planar algebra compositions. Hence, this condition is satisfied by the Khovanov complex of all alternating tangles. Finally, in the case of 0-tangles, meaning links, our condition is equivalent to a well-known result [E. S. Lee, The support of the Khovanov's invariants for alternating links, preprint (2002), arXiv:math.GT/0201105v1.] which states that the Khovanov homology of a non-split alternating link is supported on two diagonals. Thus our condition is a generalization of Lee's theorem to the case of tangles.

scholarly journals Toric Rings and Ideals of Stable Set Polytopes

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 613
Author(s):  
Kazunori Matsuda ◽  
Hidefumi Ohsugi ◽  
Kazuki Shibata
Keyword(s):  
Gröbner Bases ◽  
Infinite Family ◽  
Grobner Bases ◽  
Stable Set ◽  
Stability Number ◽  
Toric Ideals ◽  
Toric Ideal ◽  
Stable Set Polytope ◽  
Toric Rings

In the present paper, we study the normality of the toric rings of stable set polytopes, generators of toric ideals of stable set polytopes, and their Gröbner bases via the notion of edge polytopes of finite nonsimple graphs and the results on their toric ideals. In particular, we give a criterion for the normality of the toric ring of the stable set polytope and a graph-theoretical characterization of the set of generators of the toric ideal of the stable set polytope for a graph of stability number two. As an application, we provide an infinite family of stable set polytopes whose toric ideal is generated by quadratic binomials and has no quadratic Gröbner bases.

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