Knot Polynomials Research Articles

The KNTZ trick from arborescent calculus and the structure of the differential expansion

The recently suggested KNTZ trick completed the lasting search for exclusive Racah matrices $\bar S$ and $S$ for all rectangular representations and has a potential to help in the non-rectangular case as well. This was the last lacking insight about the structure of differential expansion of (rectangularly-)colored knot polynomials for twist knots -- and the resulting success is a spectacular achievement of modern knot theory in a classical field of representation theory, which was originally thought to be a tool for knot calculus but instead appeared to be its direct beneficiary. In this paper we explain that the KNTZ ansatz is actually a suggestion to convert the arborescent evolution matrix $\bar S\bar T^2\bar S$ into triangular form ${\cal B}$ and demonstrate how this works and what is the form of the old puzzles and miracles of the differential expansions from this perspective. The main new fully result is the conjecture for the triangular matrix ${\cal B}$ in the case of non-rectangular representation $[3,1]$. This paper does not simplify any calculations, but highlights the remaining problems, which one needs to overcome in order to {\it prove} that things really work. We believe that this discussion is also useful for further steps towards non-rectangular case and the related search of the gauge-invariant arborescent vertices. As an example we formulate a puzzling, still experimentally supported conjecture, that the study of twist knots only is sufficient to describe the shape of the differential expansion for all knots.

Refined Chern–Simons theory in genus two

Reshetikhin–Turaev (a.k.a. Chern–Simons) TQFT is a functor that associates vector spaces to two-dimensional genus [Formula: see text] surfaces and linear operators to automorphisms of surfaces. The purpose of this paper is to demonstrate that there exists a Macdonald [Formula: see text]-deformation — refinement — of these operators that preserves the defining relations of the mapping class groups beyond genus 1. For this, we explicitly construct the refined TQFT representation of the genus 2 mapping class group in the case of rank one TQFT. This is a direct generalization of the original genus 1 construction of arXiv:1105.5117 opening a question that if it extends to any genus. Our construction is built upon a [Formula: see text]-deformation of the square of [Formula: see text]-6[Formula: see text] symbol of [Formula: see text], which we define using the Macdonald version of Fourier duality. This allows to compute the refined Jones polynomial for arbitrary knots in genus 2. In contrast with genus 1, the refined Jones polynomial in genus 2 does not appear to agree with the Poincare polynomial of the triply graded HOMFLY knot homology.

On the Alexander polynomial of lens space knots

Ozsváth and Szabó proved the property that coefficients of the Alexander polynomial of any lens space knot are either ±1 or 0 and the non-zero coefficients are alternating. Combining the formulas of the Alexander polynomial of lens space knots by Kadokami and Yamada, and Ichihara, Saito and Teragaito, we refine Ozsváth and Szabó's property as the existence of simple curves included in a region in R2. The curves have no end-points and include just 1-component in a region. This can be much useful for studying the distribution of non-zero coefficients of the Alexander polynomial of the lens space knot. For example, we can investigate the location of the second, third and fourth non-zero coefficients. We extract a new invariant α-index. This invariant is an important factor to determine Alexander polynomials of lens space knots. We give a criterion for a lens space surgery that the Alexander polynomial is the same as a (2,r)-torus knot and a lens space surgery with small genus and so on. As well as lens space knots in S3, we also deal with lens space knots in homology spheres whose surgery duals are simple (1,1)-knots.

Kerov functions revisited

The Schur functions play a crucial role in the modern description of HOMFLY polynomials for knots and of topological vertices in DIM-based network theories, which could merge into a unified theory still to be developed. The Macdonald functions do the same for hyperpolynomials and refined vertices, but merging appears to be more problematic. For a detailed study of this problem, more knowledge is needed about the Macdonald polynomials than is usually available. As a preparation for the discussion of the knot/vertices relation, we summarize the relevant facts and open problems about the Macdonald and, more generally, Kerov functions. Like Macdonald polynomials, they are triangular combinations of Schur functions, but orthogonal in a more general scalar product. We explain that parameters of the measure can be considered as a set of new time variables, and the Kerov functions are actually expressed through the Schur functions of these variables as well. Despite they provide an infinite-parametric extension of the Schur and Macdonald polynomials, the Kerov functions, and even the skew Kerov functions continue to satisfy the most important relations, like Cauchy summation formula, the transposition identity for reflection of the Young diagram and expression of the skew functions through the generalized Littlewood–Richardson structure constants. Since these are the properties important in most applications, one can expect that the Kerov extension exists for most of them, from the superintegrable matrix and tensor models to knot theory.

Cluster variables, ancestral triangles and Alexander polynomials

In this paper, we show that Alexander polynomials for any 2-bridge knots are specializations of cluster variables. A key tool is an ancestral triangle which appeared in both quantum topology and hyperbolic geometry in different ways.

An estimation for the ascending numbers of knots by Γ-polynomials

For a knot, the ascending number is the minimum number of crossing changes which are needed to obtain an descending diagram. We study the [Formula: see text]-polynomial of knots with a given ascending number. We give a lower bound of the ascending numbers by using [Formula: see text]-polynomials. We estimate the ascending numbers for 65 prime knots up to 10 crossings by using [Formula: see text]-polynomials, Conway polynomials, and the determinants.

-DEFORMED RATIONALS AND -CONTINUED FRACTIONS

We introduce a notion of$q$-deformed rational numbers and$q$-deformed continued fractions. A$q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the$q$-deformed Pascal identity for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the$q$-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties,$q$-deformation of the Farey graph, matrix presentations and$q$-continuants are given, as well as a relation to the Jones polynomial of rational knots.

A Fox–Milnor theorem for knots in a thickened surface

A knot in a thickened surface [Formula: see text] is a smooth embedding [Formula: see text], where [Formula: see text] is a closed, connected, orientable surface. There is a bijective correspondence between knots in [Formula: see text] and knots in [Formula: see text], so one can view the study of knots in thickened surfaces as an extension of classical knot theory. An immediate question is if other classical definitions, concepts, and results extend or generalize to the study of knots in a thickened surface. One such famous result is the Fox–Milnor Theorem, which relates the Alexander polynomials of concordant knots. We prove a Fox–Milnor Theorem for concordant knots in a thickened surface by using Milnor torsion.

Evaluating the Jones polynomial with tensor networks.

We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot is reduces to the partition function of a q-state anisotropic Potts model with complex interactions, which is defined on a planar signed graph that corresponds to the knot. For any integer q, we cast this partition function into tensor network form, which inherits the interaction graph structure of the Potts model instance, and employ fast tensor network contraction protocols to obtain the exact tensor trace and thus the value of the Jones polynomial. By sampling random knots via a grid-walk procedure and computing the full tensor trace exactly, we demonstrate numerically that the Jones polynomial can be evaluated in time that scales subexponentially with the number of crossings in the typical case. This allows us to evaluate the Jones polynomial of knots that are too complex to be treated with other available methods. Our results establish tensor network methods as a practical tool for the study of knots.

On the 2-head of the colored Jones polynomial for pretzel knots

Abstract In this paper, we prove a formula for the 2-head of the colored Jones polynomial for an infinite family of pretzel knots. Following Hall, the proof utilizes skein-theoretic techniques and a careful examination of higher order stability properties for coefficients of the colored Jones polynomial.

Random Knots in 3-Dimensional 3-Colour Percolation: Numerical Results and Conjectures

Three-dimensional three-colour percolation on a lattice made of tetrahedra is a direct generalization of two-dimensional two-colour percolation on the triangular lattice. The interfaces between one-colour clusters are made of bicolour surfaces and tricolour non-intersecting and non-self-intersecting curves. Because of the three-dimensional space, these curves describe knots and links. The present paper presents a construction of such random knots using particular boundary conditions and a numerical study of some invariants of the knots. The results are sources of precise conjectures about the limit law of the Alexander polynomial of the random knots.

Extension of KNTZ trick to non-rectangular representations

We claim that the recently discovered universal-matrix precursor for the F functions, which define the differential expansion of colored polynomials for twist and double braid knots, can be extended from rectangular to non-rectangular representations. This case is far more interesting, because it involves multiplicities and associated mysterious gauge invariance of arborescent calculus. In this paper we make the very first step – reformulate in this form the previously known formulas for the simplest non-rectangular representations [r,1] and demonstrate their drastic simplification after this reformulation.

Virtual Seifert surfaces

A virtual knot that has a homologically trivial representative [Formula: see text] in a thickened surface [Formula: see text] is said to be an almost classical (AC) knot. [Formula: see text] then bounds a Seifert surface [Formula: see text]. Seifert surfaces of AC knots are useful for computing concordance invariants and slice obstructions. However, Seifert surfaces in [Formula: see text] are difficult to construct. Here, we introduce virtual Seifert surfaces of AC knots. These are planar figures representing [Formula: see text]. An algorithm for constructing a virtual Seifert surface from a Gauss diagram is given. This is applied to computing signatures and Alexander polynomials of AC knots. A canonical genus of AC knots is also studied. It is shown to be distinct from the virtual canonical genus of Stoimenow–Tchernov–Vdovina.

Multiple trait and random regression models using linear splines for genetic evaluation of multiple breed populations

This study aimed to verify if random regression models using linear splines (RRMLS) are suitable for identifying genetic parameters in multiple-breed populations and also to investigate whether an interaction exists between the breeding value (BV) of sires and their progeny breed group. Ten populations were simulated by crossing 2 breeds with distinct genetic variance and nonzero segregation variance. To obtain the genetic parameters, 2 models were used: a multiple-trait model (MULT), in which the trait was considered distinct when evaluated in each group (1/2P1 + 1/2P2, 5/8P1 + 3/8P2, and 3/4P1 + 1/4P2), and a RRMLS with the spline polynomial knots adjusted to these same groups. The genetic parameters estimated through MULT and RRMLS did not differ from the simulated values. The correlations between BV (simulated and estimated) of animals were high and varied from 0.74 to 0.76, which indicates the efficiency of using MULT and RRMLS for predicting BV. Using field data, the traits age at first calving (AFC), first lactation length (LL), and 305-d milk yield (MY-305) from a multiple-breed population of Holstein-Gyr cattle were analyzed. The BV of animals were modeled through RRMLS with 3, 5, and 7 knots, distributed in accordance with the fraction of Holstein breed in each progeny breed group. It was verified that RRMLS with 7 knots for adjusting mean trajectories and genetic effects, with homogeneous residual variance, best fit AFC and LL. For MY-305, the best fit for mean trajectory and genetic effects was the RRMLS with 5 knots and with homogeneous residual variance. The posterior means of heritability varied from 0.21 to 0.48, 0.21 to 0.38, and 0.10 to 0.33 for AFC, LL, and MY-305, respectively. Estimates from genetic parameters obtained by using RRMLS with field data showed that this model is a useful tool for genetic evaluations of populations formed by a great number of breed groups. An interaction occurred between the BV of sires and their progeny breed group, and the genetic parameters for AFC, LL, and MY-305 traits from a multiple-breed population depend on breed composition of the progeny from which the evaluations are based.

Introduction to topological quantum computation with non-Abelian anyons

Topological quantum computers promise a fault tolerant means to perform quantum computation. Topological quantum computers use particles with exotic exchange statistics called non-Abelian anyons, and the simplest anyon model which allows for universal quantum computation by particle exchange or braiding alone is the Fibonacci anyon model. One classically hard problem that can be solved efficiently using quantum computation is finding the value of the Jones polynomial of knots at roots of unity. We aim to provide a pedagogical, self-contained, review of topological quantum computation with Fibonacci anyons, from the braiding statistics and matrices to the layout of such a computer and the compiling of braids to perform specific operations. Then we use a simulation of a topological quantum computer to explicitly demonstrate a quantum computation using Fibonacci anyons, evaluating the Jones polynomial of a selection of simple knots. In addition to simulating a modular circuit-style quantum algorithm, we also show how the magnitude of the Jones polynomial at specific points could be obtained exactly using Fibonacci or Ising anyons. Such an exact algorithm seems ideally suited for a proof of concept demonstration of a topological quantum computer.

Quantum Racah matrices up to level 3 and multicolored link invariants

This paper is a next step in the project of systematic description of colored knot and link invariants started in Mironov and Morozov (2015) and Mironov et al. (2017). In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels R 1 ⊗ R 2 ⊗ R 3 ⟶ Q with all possible Q , for | R | ≤ 3 . The calculation is made possible by use of the highest weight method. The result allows one to evaluate and investigate colored polynomials for arbitrary 3-strand knots and links and to check the corresponding eigenvalue conjecture. Explicit answers for Racah matrices and colored polynomials for 3-strand knots up to 10 crossings are available at http://knotebook.org . Using the obtained inclusive Racah matrices, we also calculated the exclusive Racah matrices with the help of trick earlier suggested in the case of knots. This method is proved to be effective and gives the exclusive Racah matrices earlier obtained by another method.

Introduction to disoriented knot theory

Abstract This paper is an introduction to disoriented knot theory, which is a generalization of the oriented knot and link diagrams and an exposition of new ideas and constructions, including the basic definitions and concepts such as disoriented knot, disoriented crossing and Reidemesiter moves for disoriented diagrams, numerical invariants such as the linking number and the complete writhe, the polynomial invariants such as the bracket polynomial, the Jones polynomial for the disoriented knots and links.

Twisted Alexander polynomials of genus one two-bridge knots

Morifuji [14] computed the twisted Alexander polynomial of twist knots for nonabelian representations. In this paper we compute the twisted Alexander polynomial and Reidemeister torsion of genus one two-bridge knots, a class of knots which includes twist knots. As an application, we give a formula for the Reidemeister torsion of the 3-manifold obtained by $\frac{1}{q}$-Dehn surgery on a genus one two-bridge knot.

Twisted Alexander polynomials of 2-bridge knots associated to dihedral representations

Let [Formula: see text] be a non-abelian semi-direct product of a cyclic group [Formula: see text] and an elementary abelian [Formula: see text]-group [Formula: see text] of order [Formula: see text], [Formula: see text] being a prime and [Formula: see text]. Suppose that the knot group [Formula: see text] of a knot [Formula: see text] in the [Formula: see text]-sphere is represented on [Formula: see text]. Then we conjectured (and later proved) that the twisted Alexander polynomial [Formula: see text] associated to [Formula: see text] is of the form: [Formula: see text], where [Formula: see text] is the Alexander polynomial of [Formula: see text] and [Formula: see text] is an integer polynomial in [Formula: see text]. In this paper, we present a proof of the following. For a [Formula: see text]-bridge knot [Formula: see text] in [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text] is written as [Formula: see text], where [Formula: see text] is the set of [Formula: see text]-bridge knots whose knot groups map on that of [Formula: see text] with [Formula: see text] odd.

State invariants of 2-bridge knots

In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon–Litherland bilinear forms associated with essential state surfaces of the 2-bridge knots. We show that the resulting invariants are well-defined and explore properties of these invariants. Finally, we realize the boundary slopes of essential surfaces as differences of signatures of the knot.