Related papers: Knots and Coxeter Groups

Similar to knots in S^3, any knot in a lens space has a grid diagram from which one can combinatorially compute all of its knot Floer homology invariants. We give an explicit description of the generators, differentials, and rational Maslov…

We classify knot traces with trisection genus at most 2. We give infinitely many knots whose traces have trisection genus 3, and infinitely many knots whose traces have trisection genus 4. We also show that there exist infinite families of…

We develop obstructions to a knot K in the 3-sphere bounding a smooth punctured Klein bottle in the 4-ball. The simplest of these is based on the linking form of the 2-fold branched cover of the 3-sphere branched over K. Stronger…

We describe an algorithm to identify a minimal set of "braid relations" which span and preserve all sets of involution words for twisted Coxeter systems of finite or affine type. We classify the cases in which adding the smallest possible…

We define a knot invariant and a 2-knot invariant from any finite categorical group. We calculate an explicit example for the Spun Trefoil.

We prove that among all right-angled Coxeter groups in hyperbolic 3-space, the group generated by reflections in the faces of a right-angled triangular bipyramid with three ideal and two finite vertices has the smallest covolume. The group…

We present new computations of approximately length-minimizing polygons with fixed thickness. These curves model the centerlines of "tight" knotted tubes with minimal length and fixed circular cross-section. Our curves approximately…

In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid…

We write down an explicit formula for the $+$ version of the Heegaard Floer homology (as an absolutely graded vector space over an arbitrary field) of the results of Dehn surgery on a knot $K$ in $S^3$ in terms of homological data derived…

The space of n-sided polygons embedded in three-space consists of a smooth manifold in which points correspond to piecewise linear or ``geometric'' knots, while paths correspond to isotopies which preserve the geometric structure of these…

It is known that any two triangulations of a compact 3-manifold are related by finite sequences of certain local transformations. We prove here an upper bound for the length of a shortest transformation sequence relating any two…

By obtaining surgery descriptions of knots which lie on the genus one fiber of the trefoil or figure eight knot, we show that these include hyperbolic knots with arbitrarily large volume. These knots admit lens space surgeries and form two…

Let $K$ be a knot type for which the quadratic term of the Conway polynomial is nontrivial, and let $\gamma: \mathbb{R}\to \mathbb{R}^3$ be an analytic $\mathbb{Z}$-periodic function with non-vanishing derivative which parameterizes a knot…

In this paper, we study topological concordance modulo local knotting, or almost-concordance, of knots in 3-manifolds $M\neq S^3$. A. Levine, Celoria (arXiv:1602.05476v4), and Friedl-Nagel-Orson-Powell (arXiv:1611.09114v2) conjecture that,…

Turaev defined a function on the first homology of a rational homology 3-sphere $Y$ as the minimal rational Seifert genus of all knots in this homology class. Ni and the first author discovered a lower bound of this function using the…

We present a new method to produce simple formulas for 1-cocycles of knots over the integers, inspired by Polyak-Viro's formulas for finite-type knot invariants. We conjecture that these formulas always represent finite-type cohomology…

We show that several torsion free 3-manifold groups are not left-orderable. Our examples are groups of cyclic branched covers of S^3 branched along links. The figure eight knot provides simple nontrivial examples. The groups arising in…

An alternating torus knot or link may be constructed from a repeating double helix after connecting its two ends. A structure with additional helices may be closed to form a non-alternating torus knot or link. Previous work has optimized…

In this paper, we determine geometric information on slope lengths of a large class of knots in the 3-sphere, based only on diagrammatical properties of the knots. In particular, we show such knots have meridian length strictly less than 4,…

We demonstrate that three maximal subgroups of infinite index in the rectangular subgroup \( K_{(2,2)} \) of the Thompson group \( F \), each containing Jones's \( 3 \)-colorable subgroup \( \mathcal{F} \), can be characterized as…