The algebraic crossing number and the braid index of knots and links

10.1142/s0218216521500139 ◽

2021 ◽

pp. 2150013

Author(s):

Hwa Jeong Lee ◽

Sungjong No ◽

Seungsang Oh

Keyword(s):

Upper Bound ◽

Crossing Number ◽

Number Formula ◽

Negami found an upper bound on the stick number [Formula: see text] of a nontrivial knot [Formula: see text] in terms of the minimal crossing number [Formula: see text]: [Formula: see text]. Huh and Oh found an improved upper bound: [Formula: see text]. Huh, No and Oh proved that [Formula: see text] for a [Formula: see text]-bridge knot or link [Formula: see text] with at least six crossings. As a sequel to this study, we present an upper bound on the stick number of Montesinos knots and links. Let [Formula: see text] be a knot or link which admits a reduced Montesinos diagram with [Formula: see text] crossings. If each rational tangle in the diagram has five or more index of the related Conway notation, then [Formula: see text]. Furthermore, if [Formula: see text] is alternating, then we can additionally reduce the upper bound by [Formula: see text].

A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR UNORIENTED VIRTUAL KNOTS AND LINKS

10.1142/s0218216509007099 ◽

2009 ◽

Vol 18 (05) ◽

pp. 625-649 ◽

Cited By ~ 5

Author(s):

YASUYUKI MIYAZAWA

Keyword(s):

Crossing Number ◽

Weighted Sum ◽

Virtual Link ◽

Polynomial Invariant ◽

Virtual Knots ◽

Virtual Links ◽

We construct a multi-variable polynomial invariant Y for unoriented virtual links as a certain weighted sum of polynomials, which are derived from virtual magnetic graphs with oriented vertices, on oriented virtual links associated with a given virtual link. We show some features of the Y-polynomial including an evaluation of the virtual crossing number of a virtual link.

Braid index bounds ropelength from below

10.1142/s0218216520500194 ◽

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.

Triple-crossing projections, moves on knots and links and their minimal diagrams

10.1142/s0218216520500157 ◽

2020 ◽

Vol 29 (04) ◽

pp. 2050015 ◽

Cited By ~ 1

Author(s):

Michał Jabłonowski ◽

Łukasz Trojanowski

Keyword(s):

Triple Point ◽

Crossing Number ◽

Systematic Method ◽

Generating Set ◽

New Type ◽

Knots And Links ◽

Minimal Generating Set

In this paper, we present a systematic method to generate prime knot and prime link minimal triple-point projections, and then classify all classical prime knots and prime links with triple-crossing number at most four. We also extend the table of known knots and links with triple-crossing number equal to five. By introducing a new type of diagrammatic move, we reduce the number of generating moves on triple-crossing diagrams, and derive a minimal generating set of moves connecting triple-crossing diagrams of the same knot.

On the Minimal Crossing Number and the Braid Index of Links

Canadian Journal of Mathematics ◽

10.4153/cjm-1993-007-x ◽

1993 ◽

Vol 45 (1) ◽

pp. 117-131 ◽

Cited By ~ 17

Author(s):

Yoshiyuki Ohyama

Keyword(s):

Bipartite Graph ◽

Crossing Number ◽

Braid Index

AbstractIn this paper we prove an inequality that involves the minimal crossing number and the braid index of links by estimating Murasugi and Przytycki’s index for a planar bipartite graph.

AN EXTENDED BRACKET POLYNOMIAL FOR VIRTUAL KNOTS AND LINKS

10.1142/s0218216509007543 ◽

2009 ◽

Vol 18 (10) ◽

pp. 1369-1422 ◽

Cited By ~ 16

Author(s):

LOUIS H. KAUFFMAN

Keyword(s):

Finite Number ◽

Crossing Number ◽

Virtual Knot ◽

Virtual Knots ◽

Polynomial Coefficients ◽

Bracket Polynomial ◽

This paper defines a new invariant of virtual knots and flat virtual knots. We study this invariant in two forms: the extended bracket invariant and the arrow polyomial. The extended bracket polynomial takes the form of a sum of virtual graphs with polynomial coefficients. The arrow polynomial is a polynomial with a finite number of variables for any given virtual knot or link. We show how the extended bracket polynomial can be used to detect non-classicality and to estimate virtual crossing number and genus for virtual knots and links.

A NOTE ON CLOSED 3-BRAIDS

Communications in Contemporary Mathematics ◽

10.1142/s0219199708003150 ◽

2008 ◽

Vol 10 (supp01) ◽

pp. 1033-1047 ◽

Cited By ~ 12

Author(s):

JOAN S. BIRMAN ◽

WILLIAM W. MENASCO

Keyword(s):

Review Article ◽

Braid Index ◽

This is a review article about knots and links of braid index 3. Its goal is to gather together, in one place, some of the tools that are special to knots and links of braid index 3, in a form that could be useful for those who have a need to calculate, and need to know precisely all the exceptional cases.

Coherent band pathways between knots and links

10.1142/s0218216515500066 ◽

2015 ◽

Vol 24 (02) ◽

pp. 1550006 ◽

Cited By ~ 7

Author(s):

Dorothy Buck ◽

Kai Ishihara

Keyword(s):

Lower Bounds ◽

Recombinant Dna ◽

Crossing Number ◽

Dna Molecules ◽

Minimum Number ◽

We categorize coherent band (aka nullification) pathways between knots and 2-component links. Additionally, we characterize the minimal coherent band pathways (with intermediates) between any two knots or 2-component links with small crossing number. We demonstrate these band surgeries for knots and links with small crossing number. We apply these results to place lower bounds on the minimum number of recombinant events separating DNA configurations, restrict the recombination pathways and determine chirality and/or orientation of the resulting recombinant DNA molecules.

Closed braids and knot holders associated to some laser dynamical systems: A pump-modulated Nd-doped Â…ber laser

JOURNAL OF ADVANCES IN MATHEMATICS ◽

10.24297/jam.v12i12.4367 ◽

2017 ◽

Vol 12 (12) ◽

pp. 6894-6900

Author(s):

Elsaued Elrifai

Keyword(s):

Dynamical Systems ◽

Fiber Laser ◽

Crossing Number ◽

Topological Invariants ◽

Control Parameters ◽

Linking Number ◽

Fibered Knots ◽

Braid Theory ◽

In this work the arising knots and links for the pump-modulated Nd-doped fiber laser is investigated. For the associated templates, some of their topological invariants, such as braid linking matrix, braid words, crossing number and linking number, are studied. It is recognized that the derived topological invariants are quietly dependent on the control parameters Using the tools of the braid theory, it is shown that pump-modulated Nd-doped fiber laser knots and links are positive fibered knots and links.

A NOTE ON THE CROSSING NUMBER AND THE BRAID INDEX FOR VIRTUAL LINKS

10.1142/s021821651000825x ◽

2010 ◽

Vol 19 (07) ◽

pp. 867-880

Author(s):

YASUSHI TAKEDA

Keyword(s):

Crossing Number ◽

Virtual Link ◽

Braid Index ◽

Classical Link ◽

Virtual Links

It is well known that any virtual link is described as the closure of a virtual braid. Therefore, we can define the virtual braid index. Ohyama proved an inequality for the crossing number and the braid index of a classical link. In this paper, we prove an analogous inequality for the (total) crossing number and the braid index of a virtual link.