scholarly journals Braids with as many full twists as strands realize the braid index

2019 ◽  
Vol 12 (4) ◽  
pp. 1069-1092 ◽  
Author(s):  
Peter Feller ◽  
Diana Hubbard
Keyword(s):  
Braid Index

Twisted torus knots T(mn + m + 1,mn + 1,mn + m + 2,−1) and T(n + 1,n,2n − 1,−1) are torus knots

2021 ◽  
pp. 2150016
Author(s):  
Sangyop Lee
Keyword(s):  
Torus Knots ◽  
Torus Knot ◽  
Braid Index

A twisted torus knot [Formula: see text] is a torus knot [Formula: see text] with [Formula: see text] adjacent strands twisted fully [Formula: see text] times. In this paper, we determine the braid index of the knot [Formula: see text] when the parameters [Formula: see text] satisfy [Formula: see text]. If the last parameter [Formula: see text] additionally satisfies [Formula: see text], then we also determine the parameters [Formula: see text] for which [Formula: see text] is a torus knot.

scholarly journals On the Braid Index of Kanenobu Knots

2015 ◽  
Vol 55 (1) ◽  
pp. 169-180 ◽  
Author(s):  
Hideo Takioka
Keyword(s):  
Braid Index

On the braid index of links with nested diagrams

1992 ◽  
Vol 111 (2) ◽  
pp. 273-281 ◽  
Author(s):  
D. A. Chalcraft
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Upper Bound ◽  
Simple Criterion ◽  
Braid Index

AbstractThe number of Seifert circuits in a diagram of a link is well known 9 to be an upper bound for the braid index of the link. The -breadth of the so-called P-polynomial 3 of the link is known 5, 2 to give a lower bound. In this paper we consider a large class of links diagrams, including all diagrams where the interior of every Seifert circuit is empty. We show that either these bounds coincide, or else the upper bound is not sharp, and we obtain a very simple criterion for distinguishing these cases.

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.

scholarly journals A FAMILY OF PSEUDO-ANOSOV BRAIDS WITH LARGE CONJUGACY INVARIANT SETS

2013 ◽  
Vol 22 (06) ◽  
pp. 1350025 ◽  
Author(s):  
BYUNG HEE AN ◽  
KI HYOUNG KO
Keyword(s):  
Invariant Sets ◽  
Braid Index

We show that there is a family of pseudo-Anosov braids independently parametrized by the braid index and the (canonical) length whose smallest conjugacy invariant sets grow exponentially in the braid index and linearly in the length.

On the Minimal Crossing Number and the Braid Index of Links

1993 ◽  
Vol 45 (1) ◽  
pp. 117-131 ◽  
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.

Seifert circles and knot polynomials

1986 ◽  
Vol 99 (1) ◽  
pp. 107-109 ◽  
Author(s):  
H. R. Morton
Keyword(s):  
Braid Index ◽  
Special Case

In this paper I shall show how certain bounds on the possible diagrams presenting a given oriented knot or link K can be found from its two-variable polynomial PK defined in [3]. The inequalities regarding exponent sum and braid index of possible representations of K by a closed braid which are proved in [5] and [2] follow as a special case.

scholarly journals A formula for the braid index of links

2010 ◽  
Vol 157 (1) ◽  
pp. 247-260 ◽  
Author(s):  
Sang Youl Lee ◽  
Myoungsoo Seo
Keyword(s):  
Braid Index
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