Related papers: The representativity of pretzel knots

We study the representation spaces $R(K;\bf{i})$ as appearing in Kronheimer and Mrowka's framed instanton knot Floer homology, for a class of pretzel knots. In particular, for pretzel knots $P(p,q,r)$ with $p, q, r$ pairwise coprime, these…

In this paper, we compute the Khovanov homology over \Q for (p,-p,q) pretzel knots for odd values of p from 3 to 15 and arbitrarily large q. We provide a conjecture for the general form of the Khovanov homology of (p,-p,q) pretzel knots.…

We provide a partial classification of the 3-strand pretzel knots $K = P(p,q,r)$ with unknotting number one. Following the classification by Kobayashi and Scharlemann-Thompson for all parameters odd, we treat the remaining families with $r$…

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.

Let K_s be a (-2,3,2s+1)-type Pretzel knot (s >= 3) and E(K_s)(p/q) be a closed manifold obtained by Dehn surgery along K_s with a slope p/q. We prove that if q>0, p/q >= 4s+7 and p is odd, then E(K_s)(p/q) cannot contain an R-covered…

For each even classical pretzel knot $P(2k_1+1,2k_2+1,2k_3)$, we determine the character variety of irreducible ${\rm SL}(2,\mathbb{C})$-representations, and clarify the steps of computing its A-polynomial.

We computed the arc index for some of the pretzel knots $K=P(-p,q,r)$ with $p,q,r\ge2$, $r\geq q$ and at most one of $p,q,r$ is even. If $q=2$, then the arc index $\alpha(K)$ equals the minimal crossing number $c(K)$. If $p\ge3$ and $q=3$,…

We prove that an infinite family of three-strand pretzel knots is not squeezed. In particular, we show that $P(4, -3, 5)$ is not squeezed. This answers a question posed by Lewark (2024). Our proof is obtained by comparing the Rasmussen…

We show that the SL(2,C)-character variety of the (-2,3,n) pretzel knot consists of two (respectively three) algebraic curves when 3 does not divide n (respectively 3 divides n) and give an explicit calculation of the Culler-Shalen…

We construct a 1-parameter family of $\mathrm{SL}_2(\mathbf{R})$ representations of the pretzel knot $P(-2,3,7)$. As a consequence, we conclude that Dehn surgeries on this knot are left-orderable for all rational surgery slopes less than 6.…

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…

We determine the smooth concordance order of the 3-stranded pretzel knots P(p,q,r) with p,q,r odd. We show that each one of finite order is, in fact, ribbon, thereby proving the slice-ribbon conjecture for this family of knots. As…

Let p and q be distinct integers greater than one. We show that the 2-component pretzel link P(p,q,-p,-q) is not slice, even though it has a ribbon mutant, by using 3-fold branched covers and an obstruction based on Donaldson's…

We prove that many pretzel knots of the form $P(2n,m,-2n\pm1,-m)$ are not topologically slice, even though their positive mutants $P(2n, -2n\pm1, m, -m)$ are ribbon. We use the sliceness obstruction of Kirk and Livingston related to the…

We show that the A-polynomial $A_n$ of the 1-parameter family of pretzel knots $K_n=(-2,3,3+2n)$ satisfies a linear recursion relation of order 4 with explicit constant coefficients and initial conditions. Our proof combines results of…

We describe a method to compute the Culler-Shalen seminorms of a knot, using the (-3,3,4) pretzel knot as an illustrative example. We deduce that the SL2(C)-character variety of this knot consists of three algebraic curves and that it…

If p/q > 18, p is odd, and $p/q \ne 37/2$, (p,q)-Dehn surgery for the (-2,3,7)-pretzel knot produces a 3-manifold without Reebless foliation.

Given a knot complement X and its p-fold cyclic cover X_p, we identify twisted polynomials associated to 1-dimensional linear representations of the fundamental group of X_p with twisted polynomials associated to related p-dimensional…

Conjecture $\mathbb{Z}$ is a knot theoretical equivalent form of the Kervaire Conjecture. We say that a knot have property $\mathbb{Z}$ if it satisfies Conjecture $\mathbb{Z}$ for that specific knot. In this work, we show that alternating…

We consider the classical pretzel knots $P(a_1, a_2, a_3)$, where $a_1, a_2, a_3$ are positive odd integers. By using continuous paths of elliptic $\mathrm{SL}_2(\mathbb R)$-representations, we show that (i) the 3-manifold obtained by…