Kauffman Polynomial Research Articles

Maximal Thurston–Bennequin numbers of alternating links

We show that the upper bound of the maximal Thurston–Bennequin number for an oriented alternating link given by the Kauffman polynomial is sharp. As an application, we confirm a question of Ferrand. We also give a formula of the maximal Thurston–Bennequin number for all two-bridge links. Finally, we introduce knot concordance invariants derived from the Thurston–Bennequin number and the Maslov number of a Legendrian knot.

ON TOPOLOGICAL CORRELATIONS IN TRIVIAL KNOTS: FROM BROWNIAN BRIDGES TO CRUMPLED GLOBULES

We study, analytically and by numerical simulations, a mathematical model of closed unknotted polymer chains. The polymers are modeled by what we call "densely packed knots", which are knot diagrams entirely covering a long rectangular lattice, with randomly generated overpasses and underpasses at the nodes of the lattice. We are interested in the probability of the "knottedness" (the "knot complexity") of densily packed knots, which we measure via the degree of their Jones–Kauffman polynomial. In particular, we find the mean complexity of "daughter knots", obtained by cutting off a part of a trivial (i.e. unknotted) "parent" densily packed knot and closig up the "open tails" (the loose ends of the cut strands). We present arguments supporting the conjecture that the knot complexity n* of a daughter knot of an unknotted parent one grows as [Formula: see text] where N is the total number of vertices on the lattice. This result gives a strong support for the conjectured "crumpled globule" structure of collapsed unknotted closed polymer chains, in which the polymer forms a system of densely packed folds, mutually separated in a broad range of scales.

VASSILIEV INVARIANTS FOR VIRTUAL LINKS, CURVES ON SURFACES AND THE JONES–KAUFFMAN POLYNOMIAL

We discuss the strong invariant of virtual links proposed in [23]. This invariant is obtained as a generalization of the Jones–Kauffman polynomial (generalized Kauffman's bracket) by adding to the sum some equivalence classes of curves in two-dimensional surfaces. Thus, the invariant is valued in the infinite-dimensional free module over Z[q,q-1]. We prove that this invariant can be decomposed into finite type Vassiliev invariant of virtual links (in Kauffman's sense); thus we present new infinite series of Vassiliev invariants. It is also proved that this invariant is strictly stronger than the Jones–Kauffman polynomial for virtual knots proposed by Kauffman. Some examples when the invariant can recognize virtual knots that can not be recognized by other invariants are given.

Topological Open String Amplitudes On Orientifolds

We study topological open string amplitudes on orientifolds without fixed planes. We determine the contributions of the untwisted and twisted sectors as well as the BPS structure of the amplitudes. We illustrate our general results in various examples involving D-branes in toric orientifolds. We perform the computations by using both the topological vertex and unoriented localization. We also present an application of our results to the BPS structure of the coloured Kauffman polynomial of knots.

LENS KNOTS, PERIODIC LINKS AND VASSILIEV INVARIANTS

In this paper, we study lens knots and periodic knots by using integral Vassiliev invariants. Knot polynomials such as the Jones, HOMFLY, Kauffman polynomials give infinitely many integral Vassiliev invariants and we get some necessary conditions for a link to be a lens knot or a periodic link by using these polynomial invariants.

Coloured Tutte polynomials and Kauffman brackets for graphs of bounded tree width

Tutte polynomials are important graph invariants with rich applications in combinatorics, topology, knot theory, coding theory and even physics. The Tutte polynomial T ( G , X , Y ) is a polynomial in Z [ X , Y ] which depends on a graph G. Computing the coefficients of T ( G , X , Y ) , and even evaluating T ( G , X , Y ) at specific points ( x , y ) is ♯ P hard by a result of Jaeger et al. (Math. Proc. Cambridge Philos. Soc. 108 (1989) 35). On the other hand, Andrzejak (Discrete Math. 190 (1998) 39–54) and Noble (Combin. Probab. Comput. 7 (1998) 307–321) have shown independently, that, if G is a graph of bounded tree width, computing T ( G , X , Y ) can be done in polynomial time. We extend this result to the signed Tutte polynomials introduced in 1989 by Kauffman and the colored Tutte polynomials introduced in 1999 by Bollobas and Riordan. This allows us to prove similar results for the Jones polynomials and Kauffman brackets for knots and links which have a signed graph presentation of bounded tree width. Jones polynomials and Kauffman polynomials are the most prominent invariants of knot theory. For alternating links, they are easily computable from the Tutte polynomials of the signed graph representing the link by a result of Thistlethwaite (1988). For general links one has to use the colored Tutte polynomial instead. Knots and links can be presented as labeled planar graphs. The tree width of a link L is defined as the tree width of its graphical presentation D ( L ) as crossing diagrams. We show that for (not necessarily alternating) knots and links of tree width at most k, even the Kauffman square bracket [ L ] introduced by Bollobas and Riordan can be computed in polynomial time. Hence, the classical Kauffman bracket 〈 L 〉 and the Jones polynomial of links of tree width at most k are computable in polynomial time. Our proof is based on, but extends considerably previous work by B. Courcelle, U. Rotics and the author. It also gives a new proof of the result for Tutte polynomials and generalizes to a wide class of polynomials defined as generating functions definable in Monadic Second Order Logic with order, but invariant under it.

The Wilson Loop Observables of Chern-Simons Theory on ℝ3 in Axial Gauge

We define the Wilson loop observables (WLOs) for pure Chern-Simons models with base manifold M=ℝ3 rigorously as infinite dimensional oscillatory integrals by exploiting an ‘‘axial gauge fixing’’ and applying certain regularization techniques like ‘‘loop-smearing’’ and ‘‘framing’’. The values of the WLOs can be computed explicitly. If the structure group G of the model considered is Abelian one obtains well-known linking number expressions for the WLOs. If G is Non-Abelian one obtains expressions which are similar but not identical to the state sum representations for the Homfly and Kauffman polynomials by Jones and Turaev.

Polynomial invariants of links in the projective space

This thesis consists of two papers about knots and links in the real projective space RP3. We present an algorithm to unknot diagrammatically knots and links in RP3. We extend two classical polynomial invariants of links in R3 to links in RP3.In the first paper, the notion of descending diagram is extended from links in R3 to nonoriented links in RP3. With the help of this extended notion, any nonoriented diagram of a projective link can be unknotted with some crossing changes. It is also shown that the notion of descending diagram cannot be extended to oriented projective links.In the second paper, Homfly and Kauffman skein modules of the real projective space are computed. Both modules are free and generated by an infinite set of links, described in the paper. This explicit computation gives an extension of the Homfly and Kauffman polynomial invariants of links in R3 to links in RP3. It is also shown that the extended Homfly and Kauffman polynomials allow to measure how far a projective link is from being affine.

MULTI-VARIABLE POLYNOMIAL INVARIANTS FOR VIRTUAL LINKS

We construct new invariant polynomials in two and multiple variables for virtual knots and links. They are defined as determinants of Alexander-like matrices whose determinants are virtual link invariants. These polynomials vanish on classical links. In some cases, they separate links that can not be separated by the Jones–Kauffman polynomial [Kau] and the polynomial proposed in [Ma3].

KAUFFMAN-LIKE POLYNOMIAL AND CURVES IN 2-SURFACES

In the present paper, we construct a virtual knot invariant with values in the free infinite-dimensional module over Z[a, a-1]. The restriction of this invariant to the set of classical knots coincides with the Jones–Kauffman polynomial. It distinguishes virtual knots stronger than the generalised Jones-Kauffman polynomial proposed in [Kau].

CHERN–SIMONS THEORY, HIDA DISTRIBUTIONS, AND STATE MODELS

In this paper we present the central ideas and results of a rigorous theory of the Chern–Simons functional integral. In particular, we show that it is possible to define the Wilson loop observables (WLOs) for pure Chern–Simons models with base manifold M = ℝ3 rigorously as infinite dimensional oscillatory integrals by exploiting an "axial gauge fixing" and applying certain regularization techniques like "loop-smearing" and "framing". The (values of the) WLOs can be computed explicitly. If the structure group G of the model is Abelian one obtains well-known linking number expressions for the WLOs. If G is Non-Abelian one obtains expressions which are similar but not identical to the state model representations for the Homfly and Kauffman polynomials given in [19, 21, 31].

The parametrized complexity of knot polynomials

We study the parametrized complexity of the knot (and link) polynomials known as Jones polynomials, Kauffman polynomials and HOMFLY polynomials. It is known that computing these polynomials is ♯ P hard in general. We look for parameters of the combinatorial presentation of knots and links which make the computation of these polynomials to be fixed parameter tractable, i.e., in the complexity class FPT. If the link is explicitly presented as a closed braid, the number of its strands is known to be such a parameter. In a generalization thereof, if the link is explicitly presented as a combination of compositions and rotations of k-tangles the link is called k-algebraic, and its algebraicity k is such a parameter. The previously known proofs that, for this parameter, the link polynomials are in FPT uses the so called skein modules, and is algebraic in its nature. Furthermore, it is not clear how to find such an algebraic presentation from a given link diagram. We look at the treewidth of two combinatorial presentation of links: the crossing diagram and its shading diagram, a signed graph. We show that the treewidth of these two presentations and the algebraicity of links are all linearly related to each other. Furthermore, we characterize the k-algebraic links using the pathwidth of the crossing diagram. Using this, we can apply algorithms for testing fixed treewidth to find k-algebraic presentations in polynomial time. From this we can conclude that also treewidth and pathwidth are parameters of link diagrams for which the knot polynomials are FPT. For the Kauffman and Jones polynomials (but not for the HOMFLY polynomials) we get also a different proof for FPT via the corresponding result for signed Tutte polynomials.

Single-Boundary Iteration for Polynomial Invariants

In an earlier paper [1] we show that if a family of oriented links obey an iterative pattern, their HOMFLY polynomials satisfy a homogeneous linear recurrence relation. This also holds for Kauffman polynomials. Here we prove the result using a different construction which in some cases yields a recurrence relation of much lower order, thereby drastically reducing the amount of computation required to obtain explicit polynomials for individual members or a general formula for the entire family.

Topological notions for Kauffman and Vogel's polynomial

In [2] Kauffman and Vogel constructed a rigid vertex regular isotopy invariant for unoriented four-valent graphs embedded in three dimensional space. It assigns to each embedded graph G a polynomial, denoted [G], in three variables, A, B and a satisfying the skein relations: [Formula: see text] and is defined in terms of a state-sum and the Dubrovnik polynomial for links. In [4] it is proved, in the case B = A- 1 and a = A, that for a planar garph G we have [G] = 2c - 1 (- A - A- 1)v, where c is the number of connected components of G and v is the number of vertices of G. In this paper we will show how we can calculate the polynomial, with the variables B = A- 1 and a = A, without resorting to the skein relation.

Topological Correlations in Trivial Knots: New Arguments in Favor of the Representation of a Crumpled Polymer Globule

We prove the representation of a fractal crumpled structure of a strongly collapsed unknotted polymer chain. In this representation, topological considerations result in the chain developing a densely packed system of folds, which are mutually segregated at all scales. We investigate topological correlations in randomly generated knots on rectangular lattices (strips) of fixed widths. We find the probability of the spontaneous formation of a trivial knot and the probability that each finite part of a trivial knot becomes a trivial knot itself after joining its ends in a natural way. The complexity of a knot is characterized by the highest degree of the Jones–Kauffman polynomial topological invariant. We show that the knot complexity is proportional to the strip length in the case of long strips. Simultaneously, the typical complexity of a “quasi-knot,” which is a part of a trivial knot, is substantially less. Our analysis shows that the latter complexity is proportional to the square root of the strip length. The results obtained clearly indicate that the topological state of any part of a trivial knot densely filling the lattice is also close to the trivial state.

The Kauffman Polynomials of Generalized Hopf Links

Following the recent work by Chan [1] and Morton and Hadji [2] on the Homflypt polynomials of some generalized Hopf links, we investigate the polynomials of generalized Hopf links. By studying the Kauffman skein module of the solid torus S1 × D2, we establish a similar skein map on the Kauffman skein module of S1 × D2 which has distinct eigenvalues. Furthermore we are able to calculate the Kauffman polynomials of some specific generalized Hopf links.

Spin Models, Association Schemes and the Nakanishi–Montesinos Conjecture

A 3-transformation of a link is a local change which replaces two strings that are three times half twisted around each other by two untwisted strings (and vice versa). The Nakanishi–Montesinos (NM) conjecture asserts that this 3-transformation can unknot any link. We introduce the notion of the NM-spin model, which gives a link invariant preserved by 3-transformation. We try to classify such spin models and determine the corresponding link invariant. It is proved that the dimension of the Bose–Mesner algebra generated by the spin model is ≤4. For dimension 1 and 2, there is no such spin model, but for dimension 3, there exists a unique one. Its link invariant is a non-trivial specialization of the Kauffman polynomial, but does not distinguish trivial links from the others, and hence cannot disprove the NM conjecture. For dimension 4, we give a family of NM-spin models. The corresponding link invariant is identified and does not distinguish trivial links from the others. Strong regularity and triple regularity of the Bose–Mesner algebra and its fusions are studied.

COEFFICIENTS OF HOMFLY POLYNOMIAL AND KAUFFMAN POLYNOMIAL ARE NOT FINITE TYPE INVARIANTS

We show that the integer-valued knot invariants appearing as the nontrivial coefficients of the HOMFLY polynomial, the Kauffman polynomial and the Q-polynomial are not of finite type.

Kauffman polynomials of some links and invariants of 3-manifolds

Kauffman bracket polynomials of the so-called generalized tree-like links are studied. An algorithm of Witten type invariants, which was defined by Blanchet and Habegger et al. of more general 3-manifolds is given.

Concordance and mutation

We provide a framework for studying the interplay between concordance and positive mutation and identify some of the basic structures relating the two. The fundamental result in understanding knot concordance is the structure theorem proved by Levine: for n>1 there is an isomorphism phi from the concordance group C_n of knotted (2n-1)-spheres in S^{2n+1} to an algebraically defined group G_{+-}; furthermore, G__{+-} is isomorphic to the infinite direct sum Z^infty direct sum Z_2^infty direct sum Z_4^infty. It was a startling consequence of the work of Casson and Gordon that in the classical case the kernel of phi on C_1 is infinitely generated. Beyond this, little has been discovered about the pair (C_1,phi). In this paper we present a new approach to studying C_1 by introducing a group, M, defined as the quotient of the set of knots by the equivalence relation generated by concordance and positive mutation, with group operation induced by connected sum. We prove there is a factorization of phi, C_1-->M-->G_-. Our main result is that both maps have infinitely generated kernels. Among geometric constructions on classical knots, the most subtle is positive mutation. Positive mutants are indistinguishable using classical abelian knot invariants as well as by such modern invariants as the Jones, Homfly or Kauffman polynomials. Distinguishing positive mutants up to concordance is a far more difficult problem; only one example has been known until now. The results in this paper provide, among other results, the first infinite families of knots that are distinct from their positive mutants, even up to concordance.