scholarly journals A BRACKET POLYNOMIAL FOR GRAPHS, III: VERTEX WEIGHTS

2011 ◽  
Vol 20 (03) ◽  
pp. 435-462 ◽  
Author(s):  
LORENZO TRALDI
Keyword(s):  
Link Diagram ◽  
Kauffman Bracket ◽  
Marked Graphs ◽  
Marked Graph ◽  
Bracket Polynomial

In earlier work the Kauffman bracket polynomial was extended to an invariant of marked graphs, i.e. looped graphs whose vertices have been partitioned into two classes (marked and not marked). The marked-graph bracket polynomial is readily modified to handle graphs with weighted vertices. We present formulas that simplify the computation of this weighted bracket for graphs that contain twin vertices or are constructed using graph composition, and we show that graph composition corresponds to the construction of a link diagram from tangles.

scholarly journals A BRACKET POLYNOMIAL FOR GRAPHS, IV: UNDIRECTED EULER CIRCUITS, GRAPH-LINKS AND MULTIPLY MARKED GRAPHS

2011 ◽  
Vol 20 (08) ◽  
pp. 1093-1128 ◽  
Author(s):  
LORENZO TRALDI
Keyword(s):  
Knot Theory ◽  
Euler System ◽  
Link Diagram ◽  
Euler Systems ◽  
Kauffman Bracket ◽  
Marked Graphs ◽  
Bracket Polynomial ◽  
The Universe

In earlier work we introduced the graph bracket polynomial of graphs with marked vertices, motivated by the fact that the Kauffman bracket of a link diagram D is determined by a looped, marked version of the interlacement graph associated to a directed Euler system of the universe graph of D. Here we extend the graph bracket to graphs whose vertices may carry different kinds of marks, and we show how multiply marked graphs encode interlacement with respect to arbitrary (undirected) Euler systems. The extended machinery brings together the earlier version and the graph-links of Ilyutko and Manturov [J. Knot Theory Ramifications18 (2009) 791–823]. The greater flexibility of the extended bracket also allows for a recursive description much simpler than that of the earlier version.

Polynomial invariants for virtual marked graphs and virtual surface-link theory I

2017 ◽  
Vol 26 (12) ◽  
pp. 1750081
Author(s):  
Sang Youl Lee
Keyword(s):  
Natural Extension ◽  
Polynomial Invariants ◽  
Kauffman Bracket ◽  
Virtual Links ◽  
Link Theory ◽  
Marked Graphs ◽  
Bracket Polynomial ◽  
The Ideal

In this paper, we introduce a notion of virtual marked graphs and their equivalence and then define polynomial invariants for virtual marked graphs using invariants for virtual links. We also formulate a way how to define the ideal coset invariants for virtual surface-links using the polynomial invariants for virtual marked graphs. Examining this theory with the Kauffman bracket polynomial, we establish a natural extension of the Kauffman bracket polynomial to virtual marked graphs and found the ideal coset invariant for virtual surface-links using the extended Kauffman bracket polynomial.

An upper bound for the breadth of the Jones polynomial

1988 ◽  
Vol 103 (3) ◽  
pp. 451-456 ◽  
Author(s):  
Morwen B. Thistlethwaite
Keyword(s):  
Simple Proof ◽  
Upper Bound ◽  
Recent Article ◽  
Jones Polynomial ◽  
Upper Bounds ◽  
The Other ◽  
Link Diagram ◽  
Kauffman Bracket ◽  
Bracket Polynomial ◽  
Connected Diagram

In the recent article [2], a kind of connected link diagram called adequate was investigated, and it was shown that the Jones polynomial is never trivial for such a diagram. Here, on the other hand, upper bounds are considered for the breadth of the Jones polynomial of an arbitrary connected diagram, thus extending some of the results of [1,4,5]. Also, in Theorem 2 below, a characterization is given of those connected, prime diagrams for which the breadth of the Jones polynomial is one less than the number of crossings; recall from [1,4,5] that the breadth equals the number of crossings if and only if that diagram is reduced alternating. The article is concluded with a simple proof, using the Jones polynomial, of W. Menasco's theorem [3] that a connected, alternating diagram cannot represent a split link. We shall work with the Kauffman bracket polynomial 〈D〉 ∈ Z[A, A−1 of a link diagram D.

scholarly journals A bridge between Conway–Coxeter friezes and rational tangles through the Kauffman bracket polynomials

2019 ◽  
Vol 28 (14) ◽  
pp. 1950083 ◽  
Author(s):  
Takeyoshi Kogiso ◽  
Michihisa Wakui
Keyword(s):  
Kauffman Bracket ◽  
Rational Tangles ◽  
Bracket Polynomial ◽  
Bracket Polynomials ◽  
Zigzag Type

In this paper, we build a bridge between Conway–Coxeter friezes (CCFs) and rational tangles through the Kauffman bracket polynomials. One can compute a Kauffman bracket polynomial attached to rational links by using CCFs. As an application, one can give a complete invariant on CCFs of zigzag-type.

ON COMPUTING KAUFFMAN BRACKET POLYNOMIAL OF MONTESINOS LINKS

2010 ◽  
Vol 19 (08) ◽  
pp. 1001-1023 ◽  
Author(s):  
XIAN'AN JIN ◽  
FUJI ZHANG
Keyword(s):  
Closed Form ◽  
Explicit Expression ◽  
Computer Algebra ◽  
Jones Polynomial ◽  
Kauffman Bracket ◽  
Jones Polynomials ◽  
Bracket Polynomial ◽  
Bracket Polynomials ◽  
Closed Form Formula

It is well known that Jones polynomial (hence, Kauffman bracket polynomial) of links is, in general, hard to compute. By now, Jones polynomials or Kauffman bracket polynomials of many link families have been computed, see [4, 7–11]. In recent years, the computer algebra (Maple) techniques were used to calculate link polynomials for various link families, see [7, 12–14]. In this paper, we try to design a maple program to calculate the explicit expression of the Kauffman bracket polynomial of Montesinos links. We first introduce a family of "ring of tangles" links, which includes Montesinos links as a special subfamily. Then, we provide a closed-form formula of Kauffman bracket polynomial for a "ring of tangles" link in terms of Kauffman bracket polynomials of the numerators and denominators of the tangles building the link. Finally, using this formula and known results on rational links, the Maple program is designed.

Knots, matroids and the Ising model

1993 ◽  
Vol 113 (1) ◽  
pp. 107-139 ◽  
Author(s):  
W. Schwärzler ◽  
D. J. A. Welsh
Keyword(s):  
Ising Model ◽  
Partition Function ◽  
Knot Theory ◽  
Tutte Polynomial ◽  
Signed Graphs ◽  
Kauffman Bracket ◽  
Link Diagrams ◽  
Bracket Polynomial

AbstractA polynomial is defined on signed matroids which contains as specializations the Kauffman bracket polynomial of knot theory, the Tutte polynomial of a matroid, the partition function of the anisotropic Ising model, the Kauffman–Murasugi polynomials of signed graphs. It leads to generalizations of a theorem of Lickorish and Thistlethwaite showing that adequate link diagrams do not represent the unknot. We also investigate semi-adequacy and the span of the bracket polynomial in this wider context.

scholarly journals Lower and upper bounds of shortest paths in reachability graphs

2004 ◽  
Vol 2004 (57) ◽  
pp. 3023-3036 ◽  
Author(s):  
P. K. Mishra
Keyword(s):  
Petri Nets ◽  
Shortest Path ◽  
Petri Net ◽  
Free Choice ◽  
Shortest Paths ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Marked Graphs ◽  
Marked Graph

We prove the following property for safe marked graphs, safe conflict-free Petri nets, and live and safe extended free-choice Petri nets. We prove the following three results. If the Petri net is a marked graph, then the length of the shortest path is at most(|T|−1)⋅|T|/2. If the Petri net is conflict free, then the length of the shortest path is at most(|T|+1)⋅|T|/2. If the petrinet is live and extended free choice, then the length of the shortest path is at most|T|⋅|T+1|⋅|T+2|/6, whereTis the set of transitions of the net.

scholarly journals On generating sets of Yoshikawa moves for marked graph diagrams of surface-links

2015 ◽  
Vol 24 (04) ◽  
pp. 1550018 ◽  
Author(s):  
Jieon Kim ◽  
Yewon Joung ◽  
Sang Youl Lee
Keyword(s):  
Finite Sequence ◽  
The Other ◽  
Link Diagram ◽  
Oriented Surface ◽  
Generating Sets ◽  
Marked Graph

A marked graph diagram is a link diagram possibly with marked 4-valent vertices. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of representing surface-links by marked graph diagrams. Specially, K. Yoshikawa suggested local moves on marked graph diagrams, nowadays called Yoshikawa moves. It is now known that two marked graph diagrams representing equivalent surface-links are related by a finite sequence of these Yoshikawa moves. In this paper, we provide some generating sets of Yoshikawa moves on marked graph diagrams representing unoriented surface-links, and also oriented surface-links. We also discuss independence of certain Yoshikawa moves from the other moves.

RESULTS ON THE CLASSIFICATION OF RATIONAL 3-TANGLES

2004 ◽  
Vol 13 (02) ◽  
pp. 175-192 ◽  
Author(s):  
HUGO CABRERA-IBARRA
Keyword(s):  
Unit Disk ◽  
Braid Group ◽  
Unit Interval ◽  
Kauffman Bracket ◽  
Bracket Polynomial

An n-string tangle (B3,T) is a 3-ball B3 which contains n properly embedded arcs T={ti}, it is called rational if there is a homeomorphism of pairs from (B3,T) to (D,P)×I where D is the unit disk, P is any set of n points in the interior of D and I is the unit interval. In this article we extend the classification of the 3-braid group, [Formula: see text], obtained by using Kauffman bracket polynomial, to other families of rational 3-tangles.

On Kauffman Brackets

1997 ◽  
Vol 06 (01) ◽  
pp. 125-148 ◽  
Author(s):  
Jun Zhu
Keyword(s):  
Upper Bound ◽  
Link Diagram ◽  
Kauffman Bracket ◽  
Minimal Diagram

An antichain is associated to each link diagram so that the highest degree of the Kauffman bracket can be determined. As an application, we show that the span of the Kauffman bracket is less than or equal to 4(n - m) for dealternator connected m-alternating diagrams and the upper bound is best possible. This completely solves a conjecture of [1]. Finally, we show that a semi-alternating diagram may be not a minimal diagram which disproves a conjecture of K. Murasugi [10].

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