Related papers: Toroidal Hitomezashi Patterns
We give a proof of a Conjecture of Walker which states that one can recover the lengths of the bars of a circular linkage from the cohomology ring of the configuration space. For a large class of length vectors, this has been shown by…
Relative self-linking and linking "numbers" for pairs of knots in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual…
Talk 1: Open problems in knot theory that everyone can try to solve. Knot theory is more than two hundred years old; the first scientists who considered knots as mathematical objects were A.Vandermonde (1771) and C.F.Gauss (1794). However,…
Ozsv\'ath and Szab\'o used the knot filtration on $\widehat{CF}(S^3)$ to define the $\tau$-invariant for knots in the 3-sphere. In this article, we generalize their construction and define a collection of $\tau$-invariants associated to a…
We conjecturally extract the triply graded Khovanov-Rozansky homology of the (m, n) torus knot from the unique finite dimensional simple representation of the rational DAHA of type A, rank n - 1, and central character m/n. The conjectural…
We classify all finite group actions on knots in the 3-sphere. By geometrization, all such actions are conjugate to actions by isometries, and so we may use orthogonal representation theory to describe three cyclic and seven dihedral…
The combinatorial approach to knot theory treats knots as diagrams modulo Reidemeister moves. Many constructions of knot invariants (e.g., index polynomials, quandle colorings, etc.) use elements of diagrams such as arcs and crossings by…
This paper introduces a new approach to finding knots and links with hidden symmetries using "hidden extensions", a class of hidden symmetries defined here. We exhibit a family of tangle complements in the ball whose boundaries have…
A classical knot is described by a one-stroke trajectory with entanglements of a string. The replica method appears as a powerful tool in statistical mechanics for a polymer or self-avoiding walk. We consider this replica N to 0 limit in…
Let $f$ be a piecewise continuous and monotonic map on the interval with at most finitely many discontinuities and turning points. In this paper we study properties about this class of maps and show its main difference from the continuous…
Determining the number of stable phase-locked solutions for locally coupled Kuramoto models is a long-standing mathematical problem with important implications in biology, condensed matter physics and electrical engineering among others. We…
This note contributes to the structure theory of abstract rigidity matroids in general dimension. In the spirit of classical matroid theory, we prove several cryptomorphic characterizations of abstract rigidity matroids (in terms of…
It is well known that the rotation number of a circle homeomorphism defined by H. Poincar\'e allows to completely understand the dynamics of such a map from the topological point of view. In this paper, we collect some results concerning…
We show the (anti)holomorphicity of smooth Kobayashi isometries of the diamond, the domain defined as $\triangle:=\{z\in\mathbb C^2:|z_1|+|z_2|<1\}$. Additionally, we discuss the problem of uniqueness of real geodesics, left inverses and…
Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmueuller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a…
We investigate, in some details, symplectic equivalence between several conformal classes of Lorentz metrics on the hyperboloid of one sheet $H^{1,1} \cong T \times T - \Delta$ and affine coadjoint orbits of the group $Diff_+(\Delta)$ of…
Starting from expressions in Connor et al. (1988) [1], we derive a one-dimensional tearing equation similar to the approximate equation obtained by Hegna and Callen (1984) [2] and by Nishimura et al (1998) [3], but for more realistic…
Electromagnetic modeling provides an interesting context to present a link between physical phenomena and homology and cohomology theories. Over the past twenty-five years, a considerable effort has been invested by the computational…
A cut ideal of a graph was introduced by Sturmfels and Sullivant. In this paper, we give a necessary and sufficient condition for toric rings associated to the cut ideal to be Strongly Koszul.
We prove a structure theorem for ergodic homological rotation sets of homeomorphisms isotopic to the identity on a closed orientable hyperbolic surface: this set is made of a finite number of pieces that are either one-dimensional or almost…