Related papers: Fibered ribbon pretzels

The authors conjectured previously that a knot is nonfibered if and only if its infinite cyclic cover has uncountably many finite covers. We prove the conjecture for a class of knots that includes all knots of genus 1, using techniques from…

We conjecture explicit evolution formulas for Khovanov polynomials for pretzel knots in some regions in the windings space. Our description is exhaustive for genera 1 and 2. As previously observed, evolution at T != -1 is not fully smooth:…

Given a thin strip of paper, tie a knot, connect the ends, and flatten into the plane. This is a physical model of a folded ribbon knot in the plane, first introduced by Louis Kauffman. We study the folded ribbonlength of these folded…

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 prove that all knots can be embedded into the Menger Sponge fractal. We prove that all Pretzel knots can be embedded into the Sierpinski Tetrahedron. Then we compare the number of iterations of each of these fractals needed to produce a…

It was recently shown that a large class of phylogenetic networks, the `labellable' networks, is in bijection with the set of `expanding' covers of finite sets. In this paper, we show how several prominent classes of phylogenetic networks…

Upper bounds on projective rigidity of each homogeneously embedded homogeneous variety are determined; and a new, invariant characterization of the Fubini forms is given.

A closer look at an example introduced by Livingston & Melvin and later studied by Miyazaki shows that a plumbing of two fibered ribbon knots (along their fiber surfaces) may be algebraically slice yet not ribbon.

Using the tensor category theory developed by Lepowsky, Zhang and the second author, we construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We…

In our previous work, we introduced a simple and explicit method for constructing a positive allowable Lefschetz fibration (PALF) from a $2$-handlebody decomposition of any given compact Stein surface. In this paper, we apply this…

We construct a new type of geometric knot theory, plumbers' knots, and solve the problems of distinguishing and enumerating such knots at a fixed level of complexity. (v2) Minor edits, added theorem 3.18. (v3) Substantial revisions,…

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…

In this article, we generalize the classification of genus one Lefschetz fibrations to genus one simplified broken Lefschetz fibrations, which have fibers of genera one and zero. We classify genus one Lefschetz fibrations over the 2-disk…

The residual torsion-free nilpotence of the commutator subgroup of a knot group has played a key role in studying the bi-orderability of knot groups. A technique developed by Mayland provides a sufficient condition for the commutator…

We prove that a simple knot in the lens space $L(p,q)$ fibers if and only if its order in homology does not divide any remainder occurring in the Euclidean algorithm applied to the pair $(p,q)$. One corollary is that if $p=m^2$ is a perfect…

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…

We provide explicit formulas for the Alexander polynomial of pretzel knots and establish several immediate corollaries, including the characterization of pretzel knots with a trivial Alexander polynomial. As an application, we construct a…

We classify 3-braid knots whose topological 4-genus coincides with their Seifert genus, using McCoy's twisting method and the Xu normal form. In addition, we give upper bounds for the topological 4-genus of positive and strongly…

We show that every $\mathbb{Z}$-torsion free knot module is realized by a ribbon 2-knot with group of geometric dimension at most 2, and give some partial results on the characterization of the knot modules of fibred ribbon 2-knots.

With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend…