On alternating closed braids

2021 ◽  
pp. 2150017
Author(s):  
María de los Angeles Guevara-Hernández ◽  
Akio Kawauchi
Keyword(s):  
Numerical Invariant ◽  
Alternating Links

We introduce a numerical invariant called the braid alternation number that measures how far a link is from being an alternating closed braid. This invariant resembles the alternation number, which was previously introduced by the second author. However, these invariants are not equal, even for alternating links. We study the relation of this invariant with others and calculate this invariant for some infinite knot families. In particular, we show arbitrarily large gaps between the braid alternation number and the alternation and unknotting numbers. Furthermore, we estimate the braid alternation number for prime knots with nine crossings or less.

PROPERTIES OF VARIETIES DETERMINED BY THE DEGREES OF PROPER HYPERSUBSTITUTIONS

2008 ◽  
Vol 01 (01) ◽  
pp. 53-68 ◽  
Author(s):  
Klaus Denecke ◽  
Rattana Srithus
Keyword(s):  
Positive Integer ◽  
Numerical Invariant

The degree of proper hypersubstitutions is a numerical invariant of a variety. The main problem related to the degree of proper hypersubstitutions is to characterize all varieties of a given type τ which have, for a given positive integer n, this n as degree of proper hypersubstitutions. In this paper we answer to this question for solid varieties and certain integers as degrees and then for not necessarily solid varieties of type τ = (n) satisfying some additional conditions.

scholarly journals Almost alternating links

1992 ◽  
Vol 46 (2) ◽  
pp. 151-165 ◽  
Author(s):  
Colin C. Adams ◽  
Jeffrey F. Brock ◽  
John Bugbee ◽  
Timothy D. Comar ◽  
Keith A. Faigin ◽  
...  
Keyword(s):  
Alternating 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.

On a Structural Property of the Groups of Alternating Links

1976 ◽  
Vol 28 (3) ◽  
pp. 568-588 ◽  
Author(s):  
E. J. Mayland ◽  
Kunio Murasugi
Keyword(s):  
Structural Property ◽  
Prime Power ◽  
Residually Finite ◽  
Alternating Links ◽  
Knot Polynomial

In this paper, we will prove, as a consequence of the main theorem,THEOREM A. (See Corollary 2.6). The group of an alternating knot, for which the leading coefficient of the knot polynomial is a prime power, is residually finite and solvable.

scholarly journals A VOLUME FORM ON THE KHOVANOV INVARIANT

2012 ◽  
Vol 21 (04) ◽  
pp. 1250032
Author(s):  
JUAN ORTIZ-NAVARRO
Keyword(s):  
Chain Complex ◽  
Khovanov Homology ◽  
Volume Form ◽  
Reidemeister Torsion ◽  
Homology Groups ◽  
Numerical Invariant ◽  
Reidemeister Moves ◽  
Invariant Volume ◽  
Knots And Links

The Reidemeister torsion construction can be applied to the chain complex used to compute the Khovanov homology of a knot or a link. This defines a volume form on Khovanov homology. The volume form transforms correctly under Reidemeister moves to give an invariant volume on the Khovanov homology. In this paper, its construction and invariance under these moves is demonstrated. Also, some examples of the invariant are presented for particular choices for the bases of homology groups to obtain a numerical invariant of knots and links. In these examples, the algebraic torsion seen in the Khovanov chain complex when homology is computed over ℤ is recovered.

scholarly journals The colored Kauffman Skein relation and the head and tail of the colored Jones polynomial

2017 ◽  
Vol 26 (03) ◽  
pp. 1741002 ◽  
Author(s):  
Mustafa Hajij
Keyword(s):  
Natural Extension ◽  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Alternating Links ◽  
Skein Relation

Using the colored Kauffman skein relation, we study the highest and lowest [Formula: see text] coefficients of the [Formula: see text] unreduced colored Jones polynomial of alternating links. This gives a natural extension of a result by Kauffman in regard with the Jones polynomial of alternating links and its highest and lowest coefficients. We also use our techniques to give a new and natural proof for the existence of the tail of the colored Jones polynomial for alternating links.

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