Related papers: The representativity of pretzel knots

In this paper, we investigate three geometrical invariants of knots, the height, the trunk and the representativity. First, we give a conterexample for the conjecture which states that the height is additive under connected sum of knots. We…

We show that nontrivial classical pretzel knots L(p,q,r) are hyperbolic with eight exceptions which are torus knots. We find Conway polynomials of n-pretzel links using a new computation tree. As applications, we compute the genera of…

Let K be a hyperbolic (-2,3,n) pretzel knot and M = S^3 K its complement. For these knots, we verify a conjecture of Reid and Walsh: there are at most three knot complements in the commensurability class of M. Indeed, if n \neq 7, we show…

We present a combinatorial method for a calculation of knot Floer homology with Z-coefficient of (1,1)-knots, and then demonstrate it for non-alternating (1,1)-knots with ten crossings and the pretzel knots of type (-2,m,n). Our…

Every torus knot can be represented as a Fourier-(1,1,2) knot which is the simplest possible Fourier representation for such a knot. This answers a question of Kauffman and confirms the conjecture made by Boocher, Daigle, Hoste and Zheng.…

We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured…

We compute the unknotting number of two infinite families of pretzel knots, $P(3,1,\dots,1,b)$ (with $b$ positive and odd and an odd number of 1s) and $P(3,3,3c)$ (with $c$ positive and odd). To do this, we extend a technique of Owens using…

In this paper, we provide an explicit construction of continuous paths of $\mathrm{SL}_2(\mathbb R)$-representations of the knot groups of $(-2,3,2n+1)$-pretzel knots. As an application, we show that the fundamental group of the…

We give a criterion for distinguishing a prime knot $K$ in $S^3$ from every other knot in $S^3$ using the finite quotients of $\pi_1(S^3\setminus K)$. Using recent work of Baldwin-Sivek, we apply this criterion to the hyperbolic knots…

We determine the ${\rm SL}(2,\mathbb{C})$-character variety for each odd classical pretzel knot $P(2k_1+1,2k_2+1,2k_3+1)$, and present a method for computing its A-polynomial.

We classify Dehn surgeries on (p,q,r) pretzel knots that result in a manifold of finite fundamental group. The only hyperbolic pretzel knots that admit non-trivial finite surgeries are (-2,3,7) and (-2,3,9). Agol and Lackenby's 6-theorem…

Greene-Jabuka and Lecuona confirmed the slice-ribbon conjecture for 3-stranded pretzel knots except for an infinite family $P(a,-a-2,-\frac{(a+1)^2}{2})$ where $a$ is an odd integer greater than $1$. Lecuona and Miller showed that…

We study certain linear representations of the knot group that induce augmentations of knot contact homology. This perspective on augmentations enhances our understanding of the relationship between the augmentation polynomial and the…

We prove that for each prime power $q$ there is an integer $n$ such that if $M$ is a $3$-connected, representable matroid with a PG$(n-1,q)$-minor and no $U_{2,q^2+1}$-minor, then $M$ is representable over GF$(q)$. We also show that for…

In this note we exhibit concrete examples of characterizing slopes for the knot $12n242$, aka the $(-2,3,7)$-pretzel knot. Although it was shown by Lackenby that every knot admits infinitely many characterizing slopes, the non-constructive…

This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la…

We consider quotients of the group algebra of the $3$-string braid group $B_3$ by $p$-th order generic polynomial relations on the elementary braids. In cases $p=2,3,4,5$ these quotient algebras are finite dimensional. We give…

We discuss the polynomial representation for long knots and elaborate on how to obtain them with a bound on degrees of the defining polynomials, for any knot-type.

We collect statistics which consist of the coefficients in the expansion of the generating polynomials that count the Kauffman states associated with certain classes of pretzel knots having n tangles, of r half-twists respectively.

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,…