Kauffman Bracket Research Articles

Parity in knot theory

In this work we study knot theories with a parity property for crossings: every crossing is declared to be even or odd according to a certain preassigned rule. If this rule satisfies a set of simple axioms related to the Reidemeister moves, then certain simple invariants solving the minimality problem can be defined, and invariant maps on the set of knots can be constructed.The most important example of a knot theory with parity is the theory of virtual knots. Using the parity property arising from Gauss diagrams we show that even a gross simplification of the theory of virtual knots, namely, the theory of free knots, admits simple and highly nontrivial invariants. This gives a solution to a problem of Turaev, who conjectured that all free knots are trivial.In this work we show that free knots are generally not invertible, and provide invariants which detect the invertibility of free knots.The passage to ordinary virtual knots allows us to strengthen known invariants (such as the Kauffman bracket) using parity considerations.We also discuss other examples of knot theories with parity.Bibliography: 27 items.

Kauffman knot polynomials in classical abelian Chern–Simons field theory

Kauffman knot polynomial invariants are discovered in classical abelian Chern–Simons field theory. A topological invariant t I ( L ) is constructed for a link L , where I is the abelian Chern–Simons action and t a formal constant. For oriented knotted vortex lines, t I satisfies the skein relations of the Kauffman R-polynomial; for un-oriented knotted lines, t I satisfies the skein relations of the Kauffman bracket polynomial. As an example the bracket polynomials of trefoil knots are computed, and the Jones polynomial is constructed from the bracket polynomial.

A BRACKET POLYNOMIAL FOR GRAPHS, II: LINKS, EULER CIRCUITS AND MARKED GRAPHS

Let D be an oriented classical or virtual link diagram with directed universe [Formula: see text]. Let C denote a set of directed Euler circuits, one in each connected component of U. There is then an associated looped interlacement graph [Formula: see text] whose construction involves very little geometric information about the way D is drawn in the plane; consequently [Formula: see text] is different from other combinatorial structures associated with classical link diagrams, like the checkerboard graph, which can be difficult to extend to arbitrary virtual links. [Formula: see text] is determined by three things: the structure of [Formula: see text] as a 2-in, 2-out digraph, the distinction between crossings that make a positive contribution to the writhe and those that make a negative contribution, and the relationship between C and the directed circuits in [Formula: see text] arising from the link components; this relationship is indicated by marking the vertices where C does not follow the incident link component(s). We introduce a bracket polynomial for arbitrary marked graphs, defined using either a formula involving matrix nullities or a recursion involving the local complement and pivot operations; the marked-graph bracket of [Formula: see text] is the same as the Kauffman bracket of D. This provides a unified combinatorial description of the Jones polynomial that applies seamlessly to both classical and non-classical virtual links.

On the calculation of the Kauffman bracket polynomial

In this paper, we present a new algorithm to evaluate the Kauffman bracket polynomial. The algorithm uses cyclic permutations to count the number of states obtained by the application of ‘ A’ and ‘ B’ type smoothings to the each crossing of the knot. We show that our algorithm can be implemented easily by computer programming.

AN ORIENTED MODEL FOR KHOVANOV HOMOLOGY

We give an alternative presentation of Khovanov homology of links. The original construction rests on the Kauffman bracket model for the Jones polynomial, and the generators for the complex are enhanced Kauffman states. Here we use an oriented sl(2) state model allowing a natural definition of the boundary operator as twisted action of morphisms belonging to a TQFT for trivalent graphs and surfaces. Functoriality in original Khovanov homology holds up to sign. Variants of Khovanov homology fixing functoriality were obtained by Clark–Morrison–Walker [7] and also by Caprau [6]. Our construction is similar to those variants. Here we work over integers, while the previous constructions were over gaussian integers.

ORIENTED STATE MODEL OF THE JONES POLYNOMIAL AND ITS CONNECTION TO THE DICHROMATIC POLYNOMIAL

It is well known that Kauffman constructed a state model of the Jones polynomial based on unoriented link diagrams. In his approach, in order to obtain Jones polynomial one needs to calculate both the writhe and the Kauffman bracket. Stimulated by a paper of Altintas (An oriented state model for the Jones polynomial and its applications to alternating links, Appl. Math. Comput.194 (2007) 168–178), in this paper we present a state sum model based on oriented link diagrams. In our approach, we succeed in adding the writhe to the state sum model and need not to compute the writher any more. We further show that, via our state sum model, Jones polynomial of any link (alternating or not) is a special parametrization of the dichromatic polynomial of a weighted graph with two different edge weights.

A BRACKET POLYNOMIAL FOR GRAPHS, I

A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeister equivalence to an equivalence relation on looped graphs. The graph bracket polynomial can be defined recursively using the same pivot and local complementation operations used to define the interlace polynomial, and it gives rise to a graph Jones polynomial VG(t) that is invariant under the graph Reidemeister moves.

KHOVANOV HOMOLOGY AND THE TWIST NUMBER OF ALTERNATING KNOTS

In A Volumish Theorem for the Jones Polynomial, by O. Dasbach and X. S. Lin, it was shown that the sum of the absolute value of the second and penultimate coefficient of the Jones polynomial of an alternating knot is equal to the twist number of the knot. Here we give a new proof of this result using Khovanov homology. The proof is by induction on the number of crossings using the long exact sequence in Khovanov homology corresponding to the Kauffman bracket skein relation.

Relative Tutte Polynomials for Coloured Graphs and Virtual Knot Theory

We introduce the concept of a relative Tutte polynomial of coloured graphs. We show that this relative Tutte polynomial can be computed in a way similar to the classical spanning tree expansion used by Tutte in his original paper on this subject. We then apply the relative Tutte polynomial to virtual knot theory. More specifically, we show that the Kauffman bracket polynomial (and hence the Jones polynomial) of a virtual knot can be computed from the relative Tutte polynomial of its face (Tait) graph with some suitable variable substitutions. Our method offers an alternative to the ribbon graph approach, using the face graph obtained from the virtual link diagram directly.

AN EXTENDED BRACKET POLYNOMIAL FOR VIRTUAL KNOTS AND LINKS

This paper defines a new invariant of virtual knots and flat virtual knots. We study this invariant in two forms: the extended bracket invariant and the arrow polyomial. The extended bracket polynomial takes the form of a sum of virtual graphs with polynomial coefficients. The arrow polynomial is a polynomial with a finite number of variables for any given virtual knot or link. We show how the extended bracket polynomial can be used to detect non-classicality and to estimate virtual crossing number and genus for virtual knots and links.

FUNCTORS EXTENDING THE KAUFFMAN BRACKET

In this article we extend evaluations of the Kauffman bracket on regular isotopy classes of knots and links to a variety of functors defined on the category [Formula: see text] of framed tangles. We show that many such functors exist, and that they correspond up to equivalence to bilinear forms on free, finitely-generated modules over commutative rings R.

The multivariate signed Bollobás–Riordan polynomial

We generalise the signed Bollobás–Riordan polynomial of S. Chmutov and I. Pak [S. Chmutov, I. Pak, The Kauffman bracket of virtual links and the Bollobás–Riordan polynomial, Mos. Math. J. 7(3) (2007), 409–418] to a multivariate signed polynomial Z and study its properties. We prove the invariance of Z under the recently defined partial duality of S. Chmutov [S. Chmutov, Generalized duality for graphs on surfaces and the signed Bollobás–Riordan polynomial, J. Combin. Theory, Ser. B 99(3) (2009), 617–638. arXiv:0711.3490, doi:10.1016/j.jctb.2008.09.007] and show that the duality transformation of the multivariate Tutte polynomial is a direct consequence of it.

KBSM of the product of a disk with two holes and [formula omitted

We introduce diagrams and Reidemeister moves for links in F × S 1 , where F is an orientable surface. Using these diagrams we compute (in a new way) the Kauffman Bracket Skein Modules (KBSM) for D 2 × S 1 and A × S 1 , where D 2 is a disk and A is an annulus. Moreover, we also find the KBSM for the F 0 , 3 × S 1 , where F 0 , 3 denotes a disk with two holes, and thus show that the module is free.

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.

Coherence measures for heralded single-photon sources

We show that the Mobius function mu(n) is strongly asymptotically orthogonal to any polynomial nilsequence n -> F(g(n)L). Here, G is a simply-connected nilpotent Lie group with a discrete and cocompact subgroup L (so G/L is a nilmanifold), g : Z -> G is a polynomial sequence and F: G/L -> R is a Lipschitz function. More precisely, we show that the inner product of mu(n) with F(g(n)L) over {1,...,N} is bounded by 1/log^A N, for all A > 0. In particular, this implies the Mobius and Nilsequence conjecture MN(s) from our earlier paper "Linear equations in primes" for every positive integer s. This is one of two major ingredients in our programme, outlined in that paper, to establish a large number of cases of the generalised Hardy-Littlewood conjecture, which predicts how often a collection \psi_1,...,\psi_t : Z^d -> Z of linear forms all take prime values. The proof is a relatively quick application of the results in our recent companion paper on the distribution of polynomial orbits on nilmanifolds. We give some applications of our main theorem. We show, for example, that the Mobius function is uncorrelated with any bracket polynomial. We also obtain a result about the distribution of nilsequences n -> a^nxL as n ranges only over the primes.

AN APPLICATION OF TQFT: DETERMINING THE GIRTH OF A KNOT

A knot diagram can be divided by a circle into two parts, such that each part can be coded by a planar tree with integer weights on its edges. A half of the number of intersection points of this circle with the knot diagram is called the girth. The girth of a knot is the minimal girth of all diagrams of this knot. The girth of a knot minus one is an upper bound of the Heegaard genus of the 2-fold branched covering of that knot. We will use Topological Quantum Field Theory (TQFT) coming from the Kauffman bracket to determine the girth of some knots. Consequently, our method can be used to determine the Heegaard genus of the 2-fold branched covering of some knots.

CLASSIFICATION OF CLOSED VIRTUAL 2-BRAIDS

We classify closed virtual 2-braids completely as virtual links. For the proof, we use surface bracket polynomial due to Dye and Kauffman, Kuperberg's theorem which states existence and uniqueness of a minimal realization for a virtual link, and a subgroup of the 3-braid group.

Towards invariants of surfaces in $4$-space via classical link invariants

In this paper, we introduce a method to construct ambient isotopy invariants for smooth imbeddings of closed surfaces into 4-space by using hyperbolic splittings of the imbedded surfaces and an arbitrary given isotopy or regular isotopy invariant of classical knots and links in 3-space. Using this construction, adopting the Kauffman bracket polynomial as an example, we produce some invariants.

THE VIRTUAL MAGNETIC KAUFFMAN BRACKET SKEIN MODULE AND SKEIN RELATIONS FOR THE f-POLYNOMIAL

We introduce the virtual magnetic Kauffman bracket skein module, and show how to find a skein relation for the f-polynomial through the module. Furthermore, we give a unified approach to constructions of skein relations which hold under certain conditions.

DIFFERENCE AND DIFFERENTIAL EQUATIONS FOR THE COLORED JONES FUNCTION

The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by Le and the author that such sequences are q-holonomic, that is, they satisfy linear q-difference equations with coefficients Laurent polynomials in q and qn. We show from first principles that q-holonomic sequences give rise to modules over a q-Weyl ring. Frohman–Gelca–LoFaro have identified the latter ring with the ring of even functions of the quantum torus, and with the Kauffman bracket skein module of the torus. Via this identification, we study relations among the orthogonal, peripheral and recursion ideal of the colored Jones function, introduced by the above mentioned authors. In the second part of the paper, we convert the linear q-difference equations of the colored Jones function in terms of a hierarchy of linear ordinary differential equations for its loop expansion. This conversion is a version of the WKB method, and may shed some information on the problem of asymptotics of the colored Jones function of a knot.