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.

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

scholarly journals Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

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.

The Bipartite-Cylindrical Crossing Number of the Complete Bipartite Graph

2019 ◽  
Vol 36 (2) ◽  
pp. 205-220
Author(s):  
Bernardo Ábrego ◽  
Silvia Fernández-Merchant ◽  
Athena Sparks
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

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

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.

scholarly journals Seifert circles, braid index and the algebraic crossing number

2005 ◽  
Vol 153 (2-3) ◽  
pp. 303-317 ◽  
Author(s):  
Jože Malešič ◽  
Paweł Traczyk
Keyword(s):  
Crossing Number ◽  
Braid Index

scholarly journals Bisected vertex leveling of plane graphs: Braid index, arc index and delta diagrams

2018 ◽  
Vol 27 (08) ◽  
pp. 1850044
Author(s):  
Sungjong No ◽  
Seungsang Oh ◽  
Hyungkee Yoo
Keyword(s):  
Upper Bound ◽  
Plane Graph ◽  
Upper Bounds ◽  
Crossing Number ◽  
Index Formula ◽  
Plane Graphs ◽  
Planar Embedding ◽  
Braid Index ◽  
Arc Index

In this paper, we introduce a bisected vertex leveling of a plane graph. Using this planar embedding, we present elementary proofs of the well-known upper bounds in terms of the minimal crossing number on braid index [Formula: see text] and arc index [Formula: see text] for any knot or non-split link [Formula: see text], which are [Formula: see text] and [Formula: see text]. We also find a quadratic upper bound of the minimal crossing number of delta diagrams of [Formula: see text].

scholarly journals Triple crossing number and double crossing braid index

2019 ◽  
Vol 28 (02) ◽  
pp. 1950002 ◽  
Author(s):  
Daishiro Nishida
Keyword(s):  
Single Point ◽  
Infinite Family ◽  
Crossing Number ◽  
New Method ◽  
Braid Index

Traditionally, knot theorists have considered projections of knots where there are two strands meeting at every crossing. A triple crossing is a crossing where three strands meet at a single point, such that each strand bisects the crossing. In this paper we find a relationship between the triple crossing number and the double crossing braid index for unoriented links, namely [Formula: see text]. This yields a new method for determining braid indices. We find an infinite family of knots that achieve equality, which allows us to determine both the double crossing braid index and the triple crossing number of these knots.

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