Fast Straightening Algorithm for Bracket Polynomials Based on Tableau Manipulations

2018 ◽  
Author(s):  
Changpeng Shao ◽  
Hongbo Li
Keyword(s):  
Bracket Polynomials

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.

DEGREES OF THE JONES POLYNOMIALS OF CERTAIN PRETZEL LINKS

2000 ◽  
Vol 09 (07) ◽  
pp. 907-916 ◽  
Author(s):  
MASAO HARA ◽  
SEI'ICHI TANI ◽  
MAKOTO YAMAMOTO
Keyword(s):  
Nonnegative Integer ◽  
Jones Polynomial ◽  
Crossing Number ◽  
Kauffman Bracket ◽  
Jones Polynomials ◽  
Bracket Polynomials

We calculate the highest and the lowest degrees of the Kauffman bracket polynomials of certain inadequate pretzel links and show that there is a knot K such that c(K) – r _ deg VK=k for any nonnegative integer k, where c(K) is the crossing number of K and r _ deg VK is the reduced degree of the Jones polynomial VK of K.

Characterization of signed Gauss paragraphs and skew-symmetric graded matrices

2018 ◽  
Vol 27 (01) ◽  
pp. 1850002 ◽  
Author(s):  
José Gregorio Rodríguez-Nieto
Keyword(s):  
Embedded Graphs ◽  
Intersection Pairing ◽  
Virtual Knot ◽  
Virtual Knots ◽  
Virtual Links ◽  
The One ◽  
Minimal Realizations ◽  
Knot Diagrams ◽  
Bracket Polynomials

In this paper, we use theory of embedded graphs on oriented and compact [Formula: see text]-surfaces to construct minimal realizations of signed Gauss paragraphs. We prove that the genus of the ambient surface of these minimal realizations can be seen as a function of the maximum number of Carter’s circles. For the case of signed Gauss words, we use a generating set of [Formula: see text], given in [G. Cairns and D. Elton, The Planarity problem for signed Gauss world, J. Knots Theor. Ramif. 2(4) (1993) 359–367], and the intersection pairing of immersed [Formula: see text]-normal curves to present a short solution of the signed Gauss word problem. We relate this solution with the one given by Cairns and Elton. Moreover, we define the join operation on signed Gauss paragraphs to produce signed Gauss words such that both can be realized on the same minimal genus [Formula: see text]-surface. We connect the characterization of signed Gauss paragraph with the recognition virtual links problem. Also we present a combinatorial algorithm to compute, in an easier way, skew-symmetric graded matrices [V. Turaev, Cobordism of knots on surfaces, J. Topol. 1(2) (2008) 285–305] for virtual knots through the concept of triplets [M. Toro and J. Rodríguez, Triplets associated to virtual knot diagrams, Rev. Integración (2011)]. Therefore, we can prove that the Kishino’s knot is not classical, moreover, we prove that the virtual knots of the family [Formula: see text] given in [H. A. Dye, Virtual knots undetected by [Formula: see text] and [Formula: see text]-strand bracket polynomials, Topol. Appl. 153 (2005) 141–160] are not classical knots.

AKUTZU-WADATI LINK POLYNOMIALS FROM FEYNMAN-KAUFFMAN DIAGRAMS

1989 ◽  
Vol 04 (13) ◽  
pp. 3351-3373 ◽  
Author(s):  
MO-LIN GE ◽  
LU-YU WANG ◽  
KANG XUE ◽  
YONG-SHI WU
Keyword(s):  
Knot Theory ◽  
Jones Polynomial ◽  
Feynman Diagrams ◽  
Cpt Symmetry ◽  
Group Representations ◽  
Reidemeister Moves ◽  
S Matrix ◽  
Ambient Isotopy ◽  
Definition Of ◽  
Bracket Polynomials

By employing techniques familiar to particle physicists, we develop Kauffman’s state model for the Jones polynomial, which uses diagrams looking like Feynman diagrams for scattering, into a systematic, diagrammatic approach to new link polynomials. We systematize the ansatz for S matrix by symmetry considerations and find a natural interpretation for CPT symmetry in the context of knot theory. The invariance under Reidemeister moves of type III, II and I can be imposed diagrammatically step by step, and one obtains successively braid group representations, regular isotopy and ambient isotopy invariants from Kauffman’s bracket polynomials. This procedure is explicitiy carried out for the N=3 and 4 cases. N being the number of particle labels (or charges). With appropriate symmetry ansatz and with annihilation and creation included in the S matrix, we have obtained link polynomials which generalize the definition of the Akutzu-Wadati polynomials from closed braids to any oriented knots or links with explicit invariance under Reidemeister moves.

A NOTE ON THE UTILITY OF KAUFFMAN'S STATE SUMMATION FORMULA FOR THE BRACKET POLYNOMIAL

2002 ◽  
Vol 11 (01) ◽  
pp. 13-79
Author(s):  
PAUL ISIHARA ◽  
ANDREA KOENIGSBERG
Keyword(s):  
Jones Polynomial ◽  
Summation Formula ◽  
The State ◽  
Special Method ◽  
Symbolic Algebra ◽  
Crucial Information ◽  
Bracket Polynomial ◽  
Bracket Polynomials ◽  
Knots And Links ◽  
Insight Into

Kauffman's state summation formula is especially useful for computing the bracket polynomials of projection diagrams which are related by smoothings or crossing changes. This facilitates the writing of a symbolic algebra program which computes the normalized bracket polynomials and frequencies of knots and links whose projection diagrams result from a given knot's oriented projection diagram either by crossing changes or by orientation preserving smoothings called natural smoothings. These frequencies provide insight into the unknotting game (and similar resultant games) whose object is to specify crossing changes or natural smoothings that will transform a given projection diagram of a knot into a projection diagram representing an unknot (or some other specified knot or link). The practical utility of the state summation formula is greatly enhanced by means of diagrams for closed tangle sums. These diagrams offer a special cost-reducing method to obtain crucial information needed to compute the state summation formula. This special method also gives insight into why the bracket is unchanged by mutation and contributes a strategy to the enigmatic search for a non-trivial knot with Jones polynomial equal to one.

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