Related papers: Evaluating the Jones polynomial with tensor networ…
In this article, we explore a polynomial invariant for Legendrian knots which is a natural extension of Jones polynomial for (topological) knots. To this end, a new type of skein relation is introduced for the front projections of…
We derive exact relations between the Potts model partition function, or equivalently the Tutte polynomial, for a network (graph) $G$ and a network obtained from $G$ by (i) by replacing each edge (i.e., bond) of $G$ by two or more edges…
We solve the Jones conjecture, which states that the exponent sum in a minimal braid representation of a knot in S^3 is a knot invariant, by proving a generalized version of the original one. We apply contact geometry to study this problem…
The notion of chckerboard colorability for virtual links and abstract links is introduced. We study the Jones polynomials of virtual links and abstruct links. It is proved that a certain property of the Jones polynomials of classical links…
We investigate an application of crossing parity for the bracket expansion of the Jones polynomial for virtual knots. In addition we consider an application of parity for the arrow polynomial as well as for the categorifications of both…
A model of random walk on knot diagrams is used to study the Alexander polynomial and the colored Jones polynomial of knots. In this context, the inverse of the Alexander polynomial of a knot plays the role of an Ihara-Selberg zeta function…
We propose a new numerical algorithm for computing the tensor rank decomposition or canonical polyadic decomposition of higher-order tensors subject to a rank and genericity constraint. Reformulating this computational problem as a system…
Even when neural networks are widely used in a large number of applications, they are still considered as black boxes and present some difficulties for dimensioning or evaluating their prediction error. This has led to an increasing…
We give characterizations of the skein polynomial for links (as well as Jones and Alexander-Conway polynomials derivable from it), avoiding the usual "smoothing of a crossing" move. As by-products we have characterizations of these…
We construct a tensor network that delivers an unnormalized quantum state whose coefficients are the solutions to a given instance of 3SAT, an NP-complete problem. The tensor network contraction that corresponds to the norm of the state…
Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. For symmetric diagrams we develop a two-variable refinement $W_D(s,t)$ of the Jones polynomial that…
We recover the Jones polynomials of knots and links from the K-theory of a cluster C*-algebra of the sphere with two cusps. In particular, an interplay between the Chebyshev and Jones polynomials is studied.
We study q-holonomic sequences that arise as the colored Jones polynomial of knots in 3-space. The minimal-order recurrence for such a sequence is called the (non-commutative) A-polynomial of a knot. Using the "method of guessing", we…
The (Strong) Slope Conjecture relates the degree of the colored Jones polynomial of a knot to certain essential surfaces in the knot complement. We verify the Slope Conjecture and the Strong Slope Conjecture for 3-string Montesinos knots…
We describe a procedure that creates an explicit complex-valued polynomial function of three-dimensional space, whose nodal lines are the three-twist knot $5_2$. The construction generalizes a similar approach for lemniscate knots: a braid…
Knot theory is the Mathematical study of knots. In this paper we have studied the Composition of two knots. Knot theory belongs to Mathematical field of Topology, where the topological concepts such as topological spaces, homeomorphisms,…
In this paper, we study the asymptotic behavior of the colored Jones polynomials evaluated at roots of unity for a special class of knots. We show that certain limit is zero as predicted by the volume conjecture.
This paper gives a generalization of the AJL algorithm and unitary braid group representation for quantum computation of the Jones polynomial to continuous ranges of values on the unit circle of the Jones parameter. We show that our…
We present a polynomial time algorithm to approximately scale tensors of any format to arbitrary prescribed marginals (whenever possible). This unifies and generalizes a sequence of past works on matrix, operator and tensor scaling. Our…
There is a natural relationship between Jones polynomials and quantum computation. We use this relationship to show that the complexity of evaluating relative-error approximations of Jones polynomials can be used to bound the classical…