Kauffman Polynomial Research Articles

Link invariants, the chromatic polynomial and the Potts model

We study the connections between link invariants, the chromatic polynomial, geometric representations of models of statistical mechanics, and their common underlying algebraic structure. We establish a relation between several algebras and their associated combinatorial and topological quantities. In particular, we define the chromatic algebra, whose Markov trace is the chromatic polynomial $χQ$ of an associated graph, and we give applications of this new algebraic approach to the combinatorial properties of the chromatic polynomial. In statistical mechanics, this algebra occurs in the low-temperature expansion of the $Q$-state Potts model. We establish a relationship between the chromatic algebra and the $SO(3)$ Birman–Murakami–Wenzl algebra, which is an algebra-level analogue of the correspondence between the $SO(3)$ Kauffman polynomial and the chromatic polynomial.

On knot polynomials of annular surfaces and their boundary links

AbstractStoimenow and Kidwell asked the following question: letKbe a non-trivial knot, and letW(K) be a Whitehead double ofK. LetF(a,z) be the Kauffman polynomial andP(v,z) the skein polynomial. Is then always max degzPW(K) − 1 = 2 max degzFK? Here this question is rephrased in more general terms as a conjectured relation between the maximumz-degrees of the Kauffman polynomial of an annular surfaceAon the one hand, and the Rudolph polynomial on the other hand, the latter being defined as a certain Möbius transform of the skein polynomial of the boundary link ∂A. That relation is shown to hold for algebraic alternating links, thus simultaneously solving the conjecture by Kidwell and Stoimenow and a related conjecture by Tripp for this class of links. Also, in spite of the heavyweight definition of the Rudolph polynomial {K} of a linkK, the remarkably simple formula {◯}{L#M} = {L}{M} for link composition is established. This last result can be used to reduce the conjecture in question to the case of prime links.

Link polynomials derived from magnetic graphs

We introduce a graph diagram which can be regarded as a generalized link diagram. By using it, we construct two polynomial invariants for knots and links which are distinct from both the HOMFLY and the Kauffman polynomials. We show some features of the polynomials including relationships with the HOMFLY and the Kauffman polynomials.

Invariants in Low-Dimensional Topology

The workshop Invariants in low-dimensional topology , organised by Louis H. Kauffman (Chicago), Simon A. King (Jena), Vassily O. Manturov (Moscow) and Jozef Przytycki (Washington) was held May 4th–May 10th, 2008. This meeting was attended by 46 participants, including various researchers with recent PhD, one PhD student and one graduate student. The main objective of this workshop was to look intensively at the present state of the art of invariants in low dimensional topolgy, particularly invariants of knots and three dimensional manifolds. This field is rapidly growing with a remarkable influx of new ideas and techniques. This activity was originally inspired in the 1980's by the discovery of the Jones polynomial. Vaughan Jones formulated his invariant in terms of braids and von Neumann algebras and also gave a formula for it via a skein relation (a relation among knot and link diagrams) that was a clear generalization of the Conway skein relation for the classical Alexander polynomial. This led at once to the discovery by many people of a generalization called the Homflypt polynomial, and then sometime later another skein theoretic generalization – the Kauffman polynomial – and to a general skein theory for knots in three-manifolds (skein modules and skein algebras) due to Jozef Przytycki. The Jones polynomial is related to statistical mechanics. A key notion from statistical mechanics that was imported here was the idea to write an invariant in analogy to a partition function. This means that the invariant is expressed as a sum over combinatorial states related either to a link diagram or a triangulation of a three-manifold. The first example of this is the state sum for the Jones polynomial called Kauffman bracket. The state sums generalize to sums involving solutions to the Yang-Baxter Equation (originally used in statistical mechanics) and this in turn led to constructions of invariants from quantum groups and Hopf algebras. Then in the latter part of the 1980's ideas from quantum field theory (and the work of Witten) crystallized in the notion of Topological Quantum Field Theories (due to Witten and Atiyah) and corresponding invariants of knots and three-manifolds. At about the same time Vassiliev and Goussarov introduced new ideas about the space of knots and associated invariants that were quickly reformulated through the work of Birman, Lin, Kontsevich and Bar-Natan yielding deep relationships with Lie algebras and new ways to understand the previous constructions involving quantum groups. In 1996 Kauffman introduced a generalization of knot theory called virtual knot theory, which is given diagrammatically, and is equivalent to a stabilized theory of knots in thickened surfaces. At the same time there was an evolution in the concept of the fundamental group of the knot complement with Joyce's introduction of the quandle in 1979 and Matveev's independent discovery of this structure (distributive groupoids). After 2000 Fenn, Kauffman, Manturov and others generalized the quandle to a stronger invariant called the biquandle which is very important for virtual knots. In 1998 Khovanov introduced a categorification of the Jones polynomial. This means that he constructed a homology theory based on link diagrams whose graded Euler characteristic equals the original Jones polynomial. This led eventually to an explosion of significant results and new constructions. Khovanov and Rozansky found ways to categorify the Homflypt polynomial, and at this conference Rozansky announced the categorification of the Kauffman 2-variable polynomial. After Khovanov's original discovery, a categorification of the Alexander polynonmial (Knot Floer Homology) was found by Ozsvath and Szabo (2002) and eventually made purely combinatorial by Manolescu, Ozsvath and Sarkar. This categorification of the Alexander polynomial has led one of the most startling results in knot theory: The minimal genus of an orientable spanning surface for a knot, can be calculated from the Knot Floer homology, and this is a finite combinatorial matter in terms of a diagram for the knot. This conference collected many active researchers in this field with talks on all of the topics mentioned above. We shall now mention the main themes of the workshop and the participants who were influential or gave talks on these themes: In our opinion, many of the results in the conference were new and significant. Having the conference at the Mathematisches Forschungsinstitute enabled the participants not only to listen to new results but also to the opportunity to converse and do mathematics together. The conference was dedicated to the 60th birthday of Oleg Yannovich Viro.

On Colored Turaev-Viro Invariants for Links in Arbitrary 3-Manifolds

We consider certain invariants of links in 3-manifolds, obtained by a specialization of the Turaev-Viro invariants of 3-manifolds, that we call colored Turaev-Viro invariants. Their construction is based on a presentation of a pair (M, L), where M is a closed oriented 3-manifold, and is an oriented link, by a triangulation of M such that each component of L is an edge. We analyze some basic properties of these invariants, including the behavior under connected sums of pairs away and along links. These properties allow us to provide examples of links in having the same HOMFLY polynomial and the same Kauffman polynomial but distinct Turaev-Viro invariants, and similar examples for the Alexander polynomial. We also investigate the relations between the Turaev-Viro invariants of (M, L) and those of , showing that they are sometimes, but not always, determined by each other, and discuss some relations with the Witten-Reshetikhin-Turaev invariants and the Jones polynomial.

CHIRALITY VS. HOMFLY AND KAUFFMAN POLYNOMIALS

We exhibit an infinite family of knots that are detected chiral by the Kauffman polynomial but not by the HOMFLY polynomial.

A MULTI-VARIABLE POLYNOMIAL INVARIANT FOR VIRTUAL KNOTS AND LINKS

We construct a multi-variable polynomial invariant for virtual knots and links via the concept of a decorated virtual magnetic graph diagram. The invariant is a generalization of the Jones–Kauffman polynomial for virtual knots and links. We show some features of the invariant including an evaluation of the virtual crossing number of a virtual knot or link.

On a Background of the Existence of Multi-variable Link Invariants

Abstract. We present a quantum theorical background of the existence of multi-variable link invariants, for example the Kauffman polynomial, by observing the quan-tum (sl(2,C),ad)-invariant from the Kontsevich invariant point of view. The backgroundimplies that the Kauffman polynomial can be studied by using the sl(N,C)-skein theorysimilar to the Jones polynomial and the HOMFLY polynomial.1. IntroductionIn 1980s–90s, many multi-variable link invariants had been successfully constructed,for example, the Λ-polynomial ([7]), the Q-polynomial ([1], [4]) and the Kauffman poly-nomial. Why was it possible? In this paper, we present a quantum theorical backgroundof the existence of the above multi-variable link invariants by observing the quantum(sl(2,C),ad)-invariant from the Kontsevich invariant point of view.According to [8], the quantum (so(N),ρ 0 )-invariant, where ρ 0 is the fundamentalrepresentation of so(N), is a specialization of the Kauffman polynomial F(L;a,z) in theLaurent polynomial ring Z[a,a

Maximal Thurston--Bennequin number of $+$adequate links

The class of +adequate links contains both alternating and positive links. Generalizing results of Tanaka (for the positive case) and Ng (for the alternating case), we construct fronts of an arbitrary +adequate link A so that the diagram has a ruling; therefore its Thurston-Bennequin number is maximal among Legendrian representatives of A. We derive consequences for the Kauffman polynomial and Khovanov homology of +adequate links.

THE KAUFFMAN POLYNOMIALS OF PRETZEL KNOTS

Let ℚ(α, s) be the field of rational functions in α, s. We compute the Kauffman polynomials of pretzel knots [1,6] using the Kauffman skein theory and linear algebra tools. We give a formula for the Kauffman polynomial of a pretzel knot such that after inputting the sequence notation of the pretzel knot, the output is its Kauffman polynomial. Our calculation can be implemented in Mathematica, Maple, Mathcad, etc.

5-MOVE EQUIVALENCE CLASSES OF LINKS AND THEIR ALGEBRAIC INVARIANTS

We start a systematic analysis of links up to 5-move equivalence. Our motivation is to develop tools which later can be used to study skein modules based on the skein relation being deformation of a 5-move (in an analogous way as the Kauffman skein module is a deformation of a 2-move, i.e. a crossing change). Our main tools are Jones and Kauffman polynomials and the fundamental group of the 2-fold branch cover of S3 along a link. We use also the fact that a 5-move is a composition of two rational ±(2, 2)-moves (i.e. [Formula: see text]-moves) and rational moves can be analyzed using the group of Fox colorings and its non-abelian version, the Burnside group of a link. One curious observation is that links related by one (2, 2)-move are not 5-move equivalent. In particular, we partially classify (up to 5-moves) 3-braids, pretzel and Montesinos links, and links up to 9 crossings.

WHERE THE LINKS–GOULD INVARIANT FIRST FAILS TO DISTINGUISH NONMUTANT PRIME KNOTS

It is known that the first two-variable Links–Gould quantum link invariant LG ≡ LG2,1 is more powerful than the HOMFLYPT and Kauffman polynomials, in that it distinguishes all prime knots (including reflections) of up to 10 crossings. Here we report investigations which greatly expand the set of evaluations of LG for prime knots. Through them, we show that the invariant is complete, modulo mutation, for all prime knots (including reflections) of up to 11 crossings, but fails to distinguish some nonmutant pairs of 12-crossing prime knots. As a byproduct, we classify the mutants within the prime knots of 11 and 12 crossings. In parallel, we learn that LG distinguishes the chirality of all chiral prime knots of at most 12 crossings. We then demonstrate that every mutation-insensitive link invariant fails to distinguish the chirality of a number of 14-crossing prime knots. This provides 14-crossing examples of chiral prime knots whose chirality is undistinguished by LG.

5-moves and Montesinos links

We describe several classes of Montesinos links up to mutation and 5-move equivalence, and obtain from this a Jones and Kauffman polynomial test for a Montesinos link.

On Cabled Knots and Vassiliev Invariants (Not) Contained in Knot Polynomials

AbstractIt is known that the Brandt–Lickorish–Millett–Ho polynomialQcontains Casson's knot invariant. Whether there are (essentially) other Vassiliev knot invariants obtainable fromQis an open problem. We show that this is not so up to degree 9. We also give the (apparently) first examples of knots not distinguished by 2-cable HOMFLY polynomials which are not mutants. Our calculations provide evidence of a negative answer to the question whether Vassiliev knot invariants of degreed≤ 10 are determined by the HOMFLY and Kauffman polynomials and their 2-cables, and for the existence of algebras of such Vassiliev invariants not isomorphic to the algebras of their weight systems.

THE COEFFICIENTS OF THE HOMFLYPT AND THE KAUFFMAN POLYNOMIALS ARE POINTWISE LIMITS OF VASSILIEV INVARIANTS

The Vassiliev conjecture states that the Vassiliev invariants are dense in the space of all numerical link invariants in the sense that any link invariant is a pointwise limit of Vassiliev invariants. In this article, we prove that the Vassiliev conjecture holds in the case of the coefficients of the HOMFLY and the Kauffman polynomials.

A relation between the LG polynomial and the Kauffman polynomial

We give an explicit formula for the fact given by Links and Gould that a one variable reduction of the LG polynomial coincides with a one variable reduction of the Kauffman polynomial. This implies that the crossing number of an adequate link may be obtained from the LG polynomial by using a result of Thistlethwaite. We also give some evaluations of the LG polynomial.

Some relations on Miyazawa's virtual knot invariant

Y. Miyazawa defined a polynomial invariant for a virtual link by using magnetic graph diagrams, which is related with the Jones–Kauffman polynomial. In this paper we show some relations of this polynomial for a virtual skein triple.

ON EXTREME COEFFICIENTS OF THE JONES–KAUFFMAN POLYNOMIAL FOR VIRTUAL LINKS

An important problem of knot theory is to find or estimate the extreme coefficients of the Jones–Kauffman polynomial for (virtual) links with a given number of classical crossings. This problem has been studied by Morton and Bae [1] and Manchón [11] for the case of classical links. It turns out that the general case can be reduced to the case when the extreme coefficient function is expressible in terms of chord diagrams (previous authors consider only d-diagrams which correspond to the classical case [9]). We find the maximal absolute values for generic chord diagrams, thus, for generic virtual knots. Also we consider the "next" coefficient of the Jones–Kauffman polynomial in terms of framed chord diagrams and find its maximal value for a given number of chords. These two functions on chord diagrams are of their own interest because there are related to the Vassiliev invariants of classical knots and J-invariants of planar curves, as mentioned in [10].

Quantum link invariant from the Lie superalgebraD2 1,α

The usual construction of link invariants from quantum groups applied to the superalgebra D_{2 1,alpha} is shown to be trivial. One can modify this construction to get a two variable invariant. Unusually, this invariant is additive with respect to connected sum or disjoint union. This invariant contains an infinity of Vassiliev invariants that are not seen by the quantum invariants coming from Lie algebras (so neither by the colored HOMFLY-PT nor by the colored Kauffman polynomials).

The Thurston-Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: The Fuchs conjecture and beyond

Author(s): Rutherford, Dan | Abstract: We show that the ungraded ruling invariants of a Legendrian link can be realized as certain coefficients of the Kauffman polynomial which are non-vanishing if and only if the upper bound for the Bennequin number given by the Kauffman polynomial is sharp. This resolves positively a conjecture of Fuchs. Using similar methods a result involving the upper bound given by the HOMFLY polynomial and 2-graded rulings is proved.