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.

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 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 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.

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.

AN INFINITE FAMILY OF KNOTS WHOSE MOSAIC NUMBER IS REALIZED IN NON-REDUCED PROJECTIONS

2013 ◽  
Vol 22 (07) ◽  
pp. 1350036 ◽  
Author(s):  
LEWIS D. LUDWIG ◽  
ERICA L. EVANS ◽  
JOSEPH S. PAAT
Keyword(s):  
Infinite Family ◽  
Crossing Number

Lomonaco and Kauffman [Quantum knots and mosaics, Quantum Inf. Process. 7(2–3) (2008) 85–115] introduced the notion of knot mosaics in their work on quantum knots. It is conjectured that knot mosaic type is a complete invariant of tame knots. In this paper, we answer a question of C. Adams by constructing an infinite family of knots whose mosaic number can be realized only when the crossing number is not. That is, there is an infinite family of non-minimal knots whose mosaic numbers are known.

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.

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.

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.

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