We describe in this talk three methods of constructing different links with the same Jones type invariant. All three can be thought as generalizations of mutation. The first combines the satellite construction with mutation. The second uses the notion of rotant, taken from the graph theory, the third, invented by Jones, transplants into knot theory the idea of the Yang-Baxter equation with the spectral parameter (idea employed by Baxter in the theory of solvable models in statistical mechanics). We extend the Jones result and relate it to Traczyk's work on rotors of links. We also show further applications of the Jones idea, e.g. to 3-string links in the solid torus. We stress the fact that ideas coming from various areas of mathematics (and theoretical physics) has been fruitfully used in knot theory, and vice versa. (This is the detailed version of the talk given at the Banach Center Colloquium, Warsaw, Poland, March 24, 1994: ``W poszukiwaniu nietrywialnego wezla z trywialnym wielomianem Jonesa: grafy i mechanika statystyczna")
The noncommutative A-ideal of a knot is a generalization of the A-polynomial, defined using Kauffman bracket skein modules. In this paper we show that any knot that has the same noncommutative A-ideal as the (2,2p+1)-torus knot has the same colored Jones polynomials. This is a consequence of the orthogonality relation, which yields a recursive relation for computing all colored Jones polynomials of the knot.
In previous joint work with Frohman and Lofaro a noncommutative generalization of the A-polynomial of a knot was introduced, consisting of a finitely generated ideal of polynomials (the noncommutative A-ideal) in the quantum plane. The present paper shows that the noncommutative A-ideal of a knot, together with finitely many of its colored Jones polynomials determines all other colored Jones polynomials. Also, for particular knots, such as the unknot and the trefoil, the noncommutative A-ideal completely determines all colored Jones polynomials.
Habiro gave principal ideals of Z[q,q^{-1}] in which certain linear combinations of the colored Jones polynomials of algebraically-split links take values. The author proved that the same linear combinations for ribbon links, boundary links and Brunnian links are contained in smaller ideals of Z[q,q^{-1}] generated by several elements. In this paper, we prove that these ideals also are principal, each generated by a product of cyclotomic polynomials.
We find families of prime knot diagrams with arbitrary extreme coefficients in their Jones polynomials. Some graph theory is presented in connection with this problem, generalizing ideas by Yongju Bae and Morton and giving a positive answer to a question in their paper.
The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by TTQ. 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 $q^n$. 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, perip…
This paper is a brief overview of recent results by the authors relating colored Jones polynomials to geometric topology. The proofs of these results appear in the papers [arXiv:1002.0256] and [arXiv:1108.3370], while this survey focuses on the main ideas and examples.
This paper is a memory of the work and influence of Vaughan Jones. It is an exposition of the remarkable breakthroughs in knot theory and low dimensional topology that were catalyzed by his work. The paper recalls the inception of the Jones polynomial and the author's discovery of the bracket polynomial model for the Jones polynomial. We then describe some of the developments in knot theory that were inspired by the Jones polynomial and involve variations and generalizations of this invariant. The paper is written in the form of a personal odyssey and with the intent to show different mathemat…
The AJ conjecture, formulated by Garoufalidis, relates the A-polynomial and the colored Jones polynomial of a knot in the 3-sphere. It has been confirmed for all torus knots, some classes of two-bridge knots and pretzel knots, and most cabled knots over torus knots. The strong AJ conjecture, formulated by Sikora, relates the A-ideal and the colored Jones polynomial of a knot. It was confirmed for all torus knots. In this paper we confirm the strong AJ conjecture for most cabled knots over torus knots.
Rosso and Jones gave a formula for the colored Jones polynomial of a torus knot, colored by an irreducible representation of a simple Lie algebra. The Rosso-Jones formula involves a plethysm function, unknown in general. We provide an explicit formula for the second plethysm of an arbitrary representation of $\fsl_3$, which allows us to give an explicit formula for the colored Jones polynomial of the trefoil, and more generally, for T(2,n) torus knots. We give two independent proofs of our plethysm formula, one of which uses the work of Carini-Remmel. Our formula for the $\fsl_3$ colored Jones…
The SL_3 colored Jones polynomial of the trefoil knot is a $q$-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this $q$-holonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, this work illustrates the applicability and computational power of the employed computer algebra methods.
The pioneering work of Jones and Kauffman unveiled a fruitful relationship between statistical mechanics and knot theory. Recently, Jones introduced two subgroups $\vec{F}$ and $\vec{T}$ of the Thompson groups $F$ and $T$, respectively, together with a procedure that associates an oriented link diagram to any element of these subgroups. Moreover, several specializations of some well-known polynomial link invariants can be seen as functions of positive type on the Thompson groups or the Jones-Thompson subgroups. One important example is provided by suitable evaluations of the Jones polynomial, …
The optimistic limit is the mathematical formulation of the classical limit which is a physical method to expect the actual limit by using saddle point method of certain potential function. The original optimistic limit of the Kashaev invariant was formulated by Yokota, and a modified formulation was suggested by the author and others. The modified version was easier to handle and more combinatorial than the original one. On the other hand, it was known that the Kashaev invariant coincides with the evaluation of the colored Jones polynomial at the certain root of unity. The optimistic limit of…
This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study of knot polynomial invariants (A- or B-adequacy), we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We show that certain coefficients of the polynomial measure how far this surface is from being a fiber in the knot complement; in particular, the surface is a fiber if and…
In this paper we give a quantum statistical interpretation for the bracket polynomial state sum <K> and for the Jones polynomial. We use this quantum mechanical interpretation to give a new quantum algorithm for computing the Jones polynomial. This algorithm is useful for its conceptual simplicity, and it applies to all values of the polynomial variable that lie on the unit circle in the complex plane. Letting C(K) denote the Hilbert space for this model, there is a natural unitary transformation U from C(K) to itself such that <K> = <F|U|F> where |F> is a sum over basis states for C(K). The q…
Biquandle brackets define invariants of classical and virtual knots and links using skein invariants of biquandle-colored knots and links. Biquandle coloring quivers categorify the biquandle counting invariant in the sense of defining quiver-valued enhancements which decategorify to the counting invariant. In this paper we unite the two ideas to define biquandle bracket quivers, providing new categorifications of biquandle brackets. In particular, our construction provides an infinite family of categorifications of the Jones polynomial and other classical skein invariants.
We develop an invariant of knots that depends on a complex parameter t, describing a left ideal in the noncommutative torus. When the parameter is set equal to -1 we recover the A-polynomial of the knot. We relate the invariant to the colored Jones polynomials of the knot.
Anstee, Przyticki and Rolfsen introduced the idea of rotants, pairs of links related by a generalised form of link mutation. We exhibit infinitely many pairs of rotants which can be distinguished by Khovanov homology, but not by the Jones polynomial.
In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-zero ideal of the Weyl algebra which is generalted (after localization) by the non-commutative A-polynomial of a knot. In that paper, it was conjectured that this polynomial (which has to do with representations of the quantum group U_q(SL_2)) specializes at q=1 to the better know…
Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every $q$-holonomic sequence of rational functions. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the knot and its parallels. The topical curve explains the relation between the AJ Conjecture and the Slope Conjecture (which relate the Jones polynomial of a knot and its parallels to the $\SL(2,\BC)$ character variety and to slopes of incompressible surfaces). Our discussion predicts that the tropical curve is dual to a Newton subdivision of the $A$-polynomial o…
We introduce a polynomial invariant of graphs on surfaces, $P_G$, generalizing the classical Tutte polynomial. Topological duality on surfaces gives rise to a natural duality result for $P_G$, analogous to the duality for the Tutte polynomial of planar graphs. This property is important from the perspective of statistical mechanics, where the Tutte polynomial is known as the partition function of the Potts model. For ribbon graphs, $P_G$ specializes to the well-known Bollobas-Riordan polynomial, and in fact the two polynomials carry equivalent information in this context. Duality is also estab…
We present in this chapter (Chapter II) the history of ideas which lead up to the development of modern knot theory. We are more detailed when pre-XX century history is reported. With more recent times we are more selective, stressing developments related to Jones type invariants of links. In the Appendix, A.Przybyszewska translation of Preface to P.Heegaard Dissertation (1898) is given.
