Pretzel Knots Research Articles

Algebraic aspects of holomorphic quantum modular forms

Matrix-valued holomorphic quantum modular forms are intricate objects associated to 3-manifolds (in particular to knot complements) that arise in successive refinements of the volume conjecture of knots and involve three holomorphic, asymptotic and arithmetic realizations. It is expected that the algebraic properties of these objects can be deduced from the algebraic properties of descendant state integrals, and we illustrate this for the case of the (-2,3,7)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-2,3,7)$$\\end{document}-pretzel knot.

Knots, Perturbative Series and Quantum Modularity

We introduce an invariant of a hyperbolic knot which is a map $\alpha\mapsto \boldsymbol{\Phi}_\alpha(h)$ from $\mathbb{Q}/\mathbb{Z}$ to matrices with entries in $\overline{\mathbb{Q}}[[h]]$ and with rows and columns indexed by the boundary parabolic ${\rm SL}_2(\mathbb{C})$ representations of the fundamental group of the knot. These matrix invariants have a rich structure: (a) their $(\sigma_0,\sigma_1)$ entry, where $\sigma_0$ is the trivial and $\sigma_1$ the geometric representation, is the power series expansion of the Kashaev invariant of the knot around the root of unity ${\rm e}^{2\pi{\rm i} \alpha}$ as an element of the Habiro ring, and the remaining entries belong to generalized Habiro rings of number fields; (b) the first column is given by the perturbative power series of Dimofte-Garoufalidis; (c) the columns of $\boldsymbol{\Phi}$ are fundamental solutions of a linear $q$-difference equation; (d) the matrix defines an ${\rm SL}_2(\mathbb{Z})$-cocycle $W_\gamma$ in matrix-valued functions on $\mathbb{Q}$ that conjecturally extends to a smooth function on $\mathbb{R}$ and even to holomorphic functions on suitable complex cut planes, lifting the factorially divergent series $\boldsymbol{\Phi}(h)$ to actual functions. The two invariants $\boldsymbol{\Phi}$ and $W_\gamma$ are related by a refined quantum modularity conjecture which we illustrate in detail for the three simplest hyperbolic knots, the $4_1$, $5_2$ and $(-2,3,7)$ pretzel knots. This paper has two sequels, one giving a different realization of our invariant as a matrix of convergent $q$-series with integer coefficients and the other studying its Habiro-like arithmetic properties in more depth.

Harer–Zagier transform of the HOMFLY–PT polynomial for families of twisted hyperbolic knots

In an attempt to generalise knot matrix models for non-torus knots, which currently remains an open problem, we derived expressions for the Harer–Zagier transform—a discrete Laplace transform—of the HOMFLY–PT polynomial for some infinite families of twisted hyperbolic knots. Among them, we found a family of pretzel knots for which, like for torus knots, the transform has a fully factorised form, while for the remaining families considered it consists of sums of factorised terms. Their zero loci show a remarkable structure, which mostly lies on the unit circle and deviates from it only in pairs.

Links all of whose cyclic‐branched covers are L‐spaces

AbstractWe show that for the pretzel knots , the ‐fold cyclic‐branched covers are L‐spaces for all . In addition, we show that the knots with are quasi‐positive and slice, answering a question of Boileau–Boyer–Gordon. We also extend results of Teragaito giving examples of two‐bridge knots with all L‐space cyclic‐branched covers to a family of two‐bridge links.

Characterizations of pretzel knots which are simple-ribbon

We give a recursive formula for the Alexander polynomials of pretzel knots with a pair of integer parameters with opposite signs. Using the formula, we characterize certain pretzel knots which are simple-ribbon.

Non-Orientable Lagrangian Fillings of Legendrian Knots

AbstractWe investigate when a Legendrian knot in the standard contact ${{\mathbb{R}}}^3$ has a non-orientable exact Lagrangian filling. We prove analogs of several results in the orientable setting, develop new combinatorial obstructions to fillability, and determine when several families of knots have such fillings. In particular, we completely determine when an alternating knot (and more generally a plus-adequate knot) is decomposably non-orientably fillable and classify the fillability of most torus and 3-strand pretzel knots. We also describe rigidity phenomena of decomposable non-orientable fillings, including finiteness of the possible normal Euler numbers of fillings and the minimisation of crosscap numbers of fillings, obtaining results which contrast in interesting ways with the smooth setting.

Residual torsion-free nilpotence, biorderability and pretzel knots

Residual torsion-free nilpotence, biorderability and pretzel knots

An improvement of the lower bound for the minimum number of link colorings by quandles

We improve the lower bound for the minimum number of colors for linear Alexander quandle colorings of a knot given in Theorem 1.2 of [L. H. Kauffman and P. Lopes, Colorings beyond Fox: The other linear Alexander quandles, Linear Algebra Appl. 548 (2018) 221–258]. We express this lower bound in terms of the degree [Formula: see text] of the reduced Alexander polynomial of the knot. We show that it is exactly [Formula: see text] for L-space knots. Then we apply these results to torus knots and Pretzel knots [Formula: see text], [Formula: see text]. We note that this lower bound can be attained for some particular knots. Furthermore, we show that Theorem 1.2 quoted above can be extended to links with more than one component.

Pretzel knots up to nine crossings

There are infinitely many pretzel links with the same Alexander polynomial (actually with trivial Alexander polynomial). By contrast, in this note we revisit the Jones polynomial of pretzel links and prove that, given a natural number S, there is only a finite number of pretzel links whose Jones polynomials have span S.More concretely, we provide an algorithm useful for deciding whether or not a given knot is pretzel. As an application we identify all the pretzel knots up to nine crossings, proving in particular that 812 is the first non-pretzel knot.

Distinguishing some genus one knots using finite quotients

We give a criterion for distinguishing a prime knot [Formula: see text] in [Formula: see text] from every other knot in [Formula: see text] using the finite quotients of [Formula: see text]. Using recent work of Baldwin–Sivek, we apply this criterion to the hyperbolic knots [Formula: see text], [Formula: see text], and the three-strand pretzel knots [Formula: see text] for every integer [Formula: see text].

Cosmetic surgery conjecture for three strand pretzel knots

The problem whether the purely cosmetic surgery conjecture holds for any knots is open. Using recent work of Hanselman, the conjecture was proved for any pretzel knots [A. I. Stipsicz and Z. Szabo, Purely cosmetic surgeries and pretzel knots (2020), arXiv:2006.06765]. In this paper, we prove the conjecture for any three strand pretzel knots in a way that does not depend on Heegaard Floer homology.

Knot Colorings: Coloring and Goeritz Matrices

Knot colorings are one of the simplest ways to distinguish knots, dating back to Reidemeister and popularized by Fox. In this mostly expository article, we discuss knot invariants like colorability, knot determinant, and number of colorings, and how these can be computed from either the coloring matrix or the Goeritz matrix. We give an elementary approach to this equivalence, without using any algebraic topology. We also compute the knot determinant and the nullity of pretzel knots with arbitrarily many twist regions.

Characterizing slopes for the -pretzel knot

AbstractIn this note, we exhibit concrete examples of characterizing slopes for the knot $12n242$ , also known as the $(-2,3,7)$ -pretzel knot. Although it was shown by Lackenby that every knot admits infinitely many characterizing slopes, the nonconstructive nature of the proof means that there are very few hyperbolic knots for which explicit examples of characterizing slopes are known.

The one-row-colored 𝔰𝔩3 Jones polynomials for pretzel links

The colored [Formula: see text] Jones polynomials [Formula: see text] are given by a link and an [Formula: see text]-irreducible representation of [Formula: see text]. In general, it is hard to calculate [Formula: see text] for an oriented link [Formula: see text]. However, we calculate the one-row-colored [Formula: see text] Jones polynomials [Formula: see text] for three-parameter families of oriented pretzel links [Formula: see text] by using Kuperberg’s linear skein theory by setting [Formula: see text]. Furthermore, we show the existence of the tails of [Formula: see text] for the alternating pretzel knots [Formula: see text].

Ribbonlength of families of folded ribbon knots

We study Kauffman's model of folded ribbon knots: knots made of a thin strip of paper folded flat in the plane. The folded ribbonlength is the length to width ratio of such a ribbon knot. We give upper bounds on the folded ribbonlength of 2-bridge, $(2,q)$ torus, twist, and pretzel knots, and these upper bounds turn out to be linear in the crossing number. We give a new way to fold $(p,q)$ torus knots and show that their folded ribbonlength is bounded above by $2p$. This means, for example, that the trefoil knot can be constructed with a folded ribbonlength of 6. We then show that any $(p,q)$ torus knot $K$ with $p\geq q>2$ has a constant $c>0$, such that the folded ribbonlength is bounded above by $c\cdot Cr(K)^{1/2}$. This provides an example of an upper bound on folded ribbonlength that is sub-linear in crossing number.

A note on the involutive invariants of certain pretzel knots

We compute the involutive knot invariants for pretzel knots of the form P(−2,m,n) for m and n odd and greater than or equal to 3.

Alternating odd pretzel knots and chirally cosmetic surgeries

A pair of surgeries on a knot is chirally cosmetic if they result in homeomorphic manifolds with opposite orientations. Using recent methods of Ichihara, Ito, and Saito, we show that, except for the (2, 5) and (2, 7)-torus knots, the genus 2 and 3 alternating odd pretzel knots do not admit any chirally cosmetic surgeries. Further, we show that for a fixed genus, at most finitely many alternating odd pretzel knots admit chirally cosmetic surgeries.

Non-slice 3-stranded pretzel knots

Greene-Jabuka and Lecuona confirmed the slice-ribbon conjecture for 3-stranded pretzel knots except for an infinite family [Formula: see text], where [Formula: see text] is an odd integer greater than [Formula: see text]. Lecuona and Miller showed that [Formula: see text] are not slice unless [Formula: see text]. In this note, we show that four-fifths of the remaining knots in the family are not slice.

Generalized torsion for knots with arbitrarily high genus

AbstractLet G be a group, and let g be a nontrivial element in G. If some nonempty finite product of conjugates of g equals the identity, then g is called a generalized torsion element. We say that a knot K has generalized torsion if $G(K) = \pi _1(S^3 - K)$ admits such an element. For a $(2, 2q+1)$ -torus knot K, we demonstrate that there are infinitely many unknots $c_n$ in $S^3$ such that p-twisting K about $c_n$ yields a twist family $\{ K_{q, n, p}\}_{p \in \mathbb {Z}}$ in which $K_{q, n, p}$ is a hyperbolic knot with generalized torsion whenever $|p|> 3$ . This gives a new infinite class of hyperbolic knots having generalized torsion. In particular, each class contains knots with arbitrarily high genus. We also show that some twisted torus knots, including the $(-2, 3, 7)$ -pretzel knot, have generalized torsion. Because generalized torsion is an obstruction for having bi-order, these knots have non-bi-orderable knot groups.

Framed instanton homology and concordance

We define two concordance invariants of knots using framed instanton homology. These invariants $\nu^\sharp$ and $\tau^\sharp$ provide bounds on slice genus and maximum self-linking number, and the latter is a concordance homomorphism which agrees in all known cases with the $\tau$ invariant in Heegaard Floer homology. We use $\nu^\sharp$ and $\tau^\sharp$ to compute the framed instanton homology of all nonzero rational Dehn surgeries on: 20 of the 35 nontrivial prime knots through 8 crossings, infinite families of twist and pretzel knots, and instanton L-space knots; and of 19 of the first 20 closed hyperbolic manifolds in the Hodgson--Weeks census. In another application, we determine when the cable of a knot is an instanton L-space knot. Finally, we discuss applications to the spectral sequence from odd Khovanov homology to the framed instanton homology of branched double covers, and to the behaviors of $\tau^\sharp$ and $\tau$ under genus-2 mutation.