EXPLICIT DESCRIPTION OF THE HOMFLY POLYNOMIALS FOR 2-BRIDGE KNOTS AND LINKS

2002 ◽  
Vol 11 (04) ◽  
pp. 565-574 ◽  
Author(s):  
SHIGEKAZU NAKABO
Keyword(s):  
Explicit Formula ◽  
Classification Problem ◽  
Explicit Description ◽  
Homfly Polynomial ◽  
Homfly Polynomials ◽  
Knots And Links

An explicit formula of the HOMFLY polynomial of 2-bridge knots and links is presented. As corollaries, some specific coefficient polynomials are described explicitly. Lastly, some examples are calculated, which are related to the classification problem of the 2-bridge knots and links by the HOMFLY polynomial.

FORMULAS FOR THE CASSON INVARIANT OF CERTAIN INTEGRAL HOMOLOGY 3-SPHERES

2009 ◽  
Vol 18 (11) ◽  
pp. 1551-1576 ◽  
Author(s):  
SANG YOUL LEE ◽  
MYOUNGSOO SEO
Keyword(s):  
Explicit Formula ◽  
Dehn Surgery ◽  
Integral Homology ◽  
Integral Matrices ◽  
Knots And Links ◽  
Special Classes

In this paper, we introduce a representation of knots and links in S3 by integral matrices and then give an explicit formula for the Casson invariant for integral homology 3-spheres obtained from S3 by Dehn surgery along the knots and links represented by the integral matrices in which either all entries are even or the entries of each row are the same odd number. As applications, we study the preimage of the Casson invariant for a given integer and also give formulas for the Casson invariants of some special classes of integral homology 3-spheres.

THE HOMFLY POLYNOMIAL FOR TORUS LINKS FROM CHERN–SIMONS GAUGE THEORY

1995 ◽  
Vol 10 (07) ◽  
pp. 1045-1089 ◽  
Author(s):  
J. M. F. LABASTIDA ◽  
M. MARIÑO
Keyword(s):  
Gauge Theory ◽  
Gauge Group ◽  
Homfly Polynomial ◽  
Fundamental Representation ◽  
Torus Knots ◽  
Polynomial Invariants ◽  
Chern Simons ◽  
Knots And Links ◽  
Torus Links

Polynomial invariants corresponding to the fundamental representation of the gauge group SU(N) are computed for arbitrary torus knots and links in the framework of Chern–Simons gauge theory making use of knot operators. As a result, a formula for the HOMFLY polynomial for arbitrary torus links is presented.

Mutant knots with symmetry

2009 ◽  
Vol 146 (1) ◽  
pp. 95-107 ◽  
Author(s):  
H. R. MORTON
Keyword(s):  
Homfly Polynomial ◽  
Homfly Polynomials

AbstractMutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general theirm-string satellites can have different Homfly polynomials form> 2. We show that, under conditions of extra symmetry on the constituent 2-tangles, the directedm-string satellites of mutants share the same Homfly polynomial form< 6 in general, and for all choices ofmwhen the satellite is based on a cable knot pattern.We give examples of mutants with extra symmetry whose Homfly polynomials of some 6-string satellites are different, by comparing their quantumsl(3) invariants.

Tori in the diffeomorphism groups of simply-connected 4-manifolds

1982 ◽  
Vol 91 (2) ◽  
pp. 305-314 ◽  
Author(s):  
Paul Melvin
Keyword(s):  
Conjugacy Class ◽  
Explicit Formula ◽  
Equivalence Class ◽  
Classification Problem ◽  
Conjugacy Classes ◽  
Weak Equivalence ◽  
Dimensional Torus ◽  
Diffeomorphism Groups ◽  
Connected Sum ◽  
Simply Connected

Let M be a closed simply-connected 4-manifold. All manifolds will be assumed smooth and oriented. The purpose of this paper is to classify up to conjugacy the topological subgroups of Diff(M) isomorphic to the 2-dimensional torus T2 (Theorem 1), and to give an explicit formula for the number of such conjugacy classes (Theorem 2). Such a conjugacy class corresponds uniquely to a weak equivalence class of effective T2-actions on M. Thus the classification problem is trivial unless M supports an effective T2-action. Orlik and Raymond showed that this happens if and only if M is a connected sum of copies of ± ;P2 and S2 × S2 (2), and so this paper is really a study of the different T2-actions on these manifolds.

scholarly journals Colored HOMFLY polynomials for the pretzel knots and links

2015 ◽  
Vol 2015 (7) ◽  
Author(s):  
A. Mironov ◽  
A. Morozov ◽  
A. Sleptsov
Keyword(s):  
Pretzel Knots ◽  
Homfly Polynomials ◽  
Knots And Links

scholarly journals COLORED HOMFLY POLYNOMIALS FROM CHERN–SIMONS THEORY

2013 ◽  
Vol 22 (13) ◽  
pp. 1350078 ◽  
Author(s):  
SATOSHI NAWATA ◽  
P. RAMADEVI ◽  
ZODINMAWIA
Keyword(s):  
Simons Theory ◽  
Theoretic Method ◽  
Two Component ◽  
6J Symbols ◽  
Multiplicity Free ◽  
Homfly Polynomials ◽  
Chern Simons ◽  
Knots And Links ◽  
Chern Simons Theory

We elaborate the Chern–Simons field theoretic method to obtain colored HOMFLY invariants of knots and links. Using multiplicity-free quantum 6j-symbols for Uq(𝔰𝔩N), we present explicit evaluations of the HOMFLY invariants colored by symmetric representations for a variety of knots, two-component links and three-component links.

scholarly journals INVARIANTS OF GENUS 2 MUTANTS

2009 ◽  
Vol 18 (10) ◽  
pp. 1423-1438 ◽  
Author(s):  
H. R. MORTON ◽  
N. RYDER
Keyword(s):  
Jones Polynomial ◽  
Homfly Polynomial ◽  
Vassiliev Invariant ◽  
Genus 2 ◽  
Homfly Polynomials

Pairs of genus 2 mutant knots can have different Homfly polynomials, for example some 3-string satellites of Conway mutant pairs. We give examples which have different Kauffman 2-variable polynomials, answering a question raised by Dunfield et al. in their study of genus 2 mutants. While pairs of genus 2 mutant knots have the same Jones polynomial, given from the Homfly polynomial by setting v = s2, we give examples whose Homfly polynomials differ when v = s3. We also give examples which differ in a Vassiliev invariant of degree 7, in contrast to satellites of Conway mutant knots.

ON THE QUANTUM COMPLEXITY OF EVALUATING THE TUTTE POLYNOMIAL

2010 ◽  
Vol 19 (06) ◽  
pp. 727-737
Author(s):  
HAMED AHMADI ◽  
PAWEL WOCJAN
Keyword(s):  
Braid Group ◽  
Quantum Computer ◽  
Tutte Polynomial ◽  
Roots Of Unity ◽  
Homfly Polynomial ◽  
Root Of Unity ◽  
Quantum Complexity ◽  
Homfly Polynomials ◽  
Alternating Links ◽  
Alternating Link

We show that the problem of approximately evaluating the Tutte polynomial of triangular graphs at the points (q, 1/q) of the Tutte plane is BQP-complete for (most) roots of unity q. We also consider circular graphs and show that the problem of approximately evaluating the Tutte polynomial of these graphs at the point (e2πi/5, e-2πi/5) is DQC1-complete and at points [Formula: see text] for some integer k is in BQP. To show that these problems can be solved by a quantum computer, we rely on the relation of the Tutte polynomial of a planar G graph with the Jones and HOMFLY polynomial of the alternating link D(G) given by the medial graph of G. In the case of our graphs the corresponding links are equal to the plat and trace closures of braids. It is known how to evaluate the Jones and HOMFLY polynomial for closures of braids. To establish the hardness results, we use the property that the images of the generators of the braid group under the irreducible Jones–Wenzl representations of the Hecke algebra have finite order. We show that for each braid b we can efficiently construct a braid [Formula: see text] such that the evaluation of the Jones and HOMFLY polynomials of their closures at a fixed root of unity leads to the same value and that the closures of [Formula: see text] are alternating links.

scholarly journals Evolution method and HOMFLY polynomials for virtual knots

2015 ◽  
Vol 30 (14) ◽  
pp. 1550074 ◽  
Author(s):  
Ludmila Bishler ◽  
Alexei Morozov ◽  
Andrey Morozov ◽  
Anton Morozov
Keyword(s):  
Infinite Series ◽  
Symmetric Representation ◽  
Virtual Knots ◽  
Topological Invariance ◽  
Definition Of ◽  
Homfly Polynomials ◽  
Knots And Links

Following the suggestion of Alexei Morozov, Andrey Morozov and Anton Morozov, Phys. Lett. B737, 48 (2014), arXiv:1407.6319, to lift the knot polynomials for virtual knots and links from Jones to HOMFLY, we apply the evolution method to calculate them for an infinite series of twist-like virtual knots and antiparallel two-strand links. Within this family one can check topological invariance and understand how differential hierarchy is modified in virtual case. This opens a way towards a definition of colored (not only cabled) knot polynomials, though problems still persist beyond the first symmetric representation.

DISTINGUISHING MUTANTS BY KNOT POLYNOMIALS

1996 ◽  
Vol 05 (02) ◽  
pp. 225-238 ◽  
Author(s):  
HUGH R. MORTON ◽  
PETER R. CROMWELL
Keyword(s):  
Explicit Calculation ◽  
Homfly Polynomial ◽  
Quantum Invariants ◽  
Quantum Invariant ◽  
Vassiliev Invariant ◽  
Simple Quantum ◽  
The Difference ◽  
Homfly Polynomials

We consider the problem of distinguishing mutant knots using invariants of their satellites. We show, by explicit calculation, that the Homfly polynomial of the 3-parallel (and hence the related quantum invariants) will distinguish some mutant pairs. Having established a condition on the colouring module which forces a quantum invariant to agree on mutants, we explain several features of the difference between the Homfly polynomials of satellites constructed from mutants using more general patterns. We illustrate this by our calculations; from these we isolate some simple quantum invariants, and a framed Vassiliev invariant of type 11, which distinguish certain mutants, including the Conway and Kinoshita-Teresaka pair.

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