Related papers: Additivity of Circular Width

The knots-quivers correspondence states that various characteristics of a knot are encoded in the corresponding quiver and the moduli space of its representations. However, this correspondence is not a bijection: more than one quiver may be…

We consider the question, asked by Friedl, Livingston and Zentner, of which sums of torus knots are concordant to alternating knots. After a brief analysis of the problem in its full generality, we focus on sums of two torus knots. We…

Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known, however sums…

We prove that for a given flat surface with conical singularities, any pair of geometric triangulations can be connected by a chain of flips.

We define a notion of concordance based on Euler characteristic, and show that it gives rise to a concordance group of links in the three-sphere, which has the concordance group of knots as a direct summand with infinitely generated…

We introduce the notion of wen knots, and prove that the set of wen knots is a proper subset of the set of extended welded knots. Furthermore we prove that the complementary subset consists of welded knots up to horizontal mirror…

We describe a way to decompose the chromatic symmetric function as a positive sum of smaller pieces. We show that these pieces are $e$-positive for cycles. Then we prove that attaching a cycle to a graph preserves the $e$-positivity of…

We show that if a graph is k-edge-connected, and we adjoin to it another graph satisfying a "contracted diameter less or equal to 2" condition, with minimal degree greater or equal to k, and some natural hypothesis on the edges connecting…

We study the geometry of hyperbolic knots that admit alternating projections on embedded surfaces in closed 3-manifolds. We show that, under mild hypothesis, their cusp area admits two sided bounds in terms of the twist number of the…

We show that any closed incompressible surface in the complement of a positive knot is algebraically non-split from the knot, positive knots cannot bound non-free incompressible Seifert surfaces and that the splitability and the primeness…

We prove that a special alternating knot does not decompose as a non-trivial band sum. This restricts concordances from special alternating knots, and we conjecture that special alternating knots are ribbon concordance minimal. We verify…

We study the geometry of interacting knotted solitons. The interaction is local and advances either as a three-body or as a four-body process, depending on the relative orientation and a degeneracy of the solitons involved. The splitting…

This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3-dimensional topology approach that if a…

In this paper we study how randomly generated knots occupy a volume of space using topological methods. To this end, we consider the evolution of the first homology of an immersed metric neighbourhood of a knot's embedding for growing…

Weakly generalised alternating knots are knots with an alternating projection onto a closed surface in a compact irreducible 3-manifold, and they share many hyperbolic geometric properties with usual alternating knots. For example, usual…

This paper provides two obstructions to small knot complements in $S^3$ admitting hidden symmetries. The first obstruction is being cyclically commensurable with another knot complement. This result provides a partial answer to a conjecture…

Every oriented closed geodesic on the modular surface has a canonically associated knot in its unit tangent bundle coming from the periodic orbit of the geodesic flow. We study the volume of the associated knot complement with respect to…

In this paper, we show that any knot group maps onto at most finitely many knot groups. This gives an affirmative answer to a conjecture of J. Simon. We also bound the diameter of a closed hyperbolic 3-manifold linearly in terms of the…

Let $K, K'$ be ribbon knottings of $n$-spheres with $1$-handles in $S^{n+2}$, $n\geq 2$. We show that if the knot quandles of these knots are isomorphic, then the ribbon knottings are stably equivalent, in the sense of Nakanishi and…

Quandle colorings and cocycle invariants are studied for composite knots, and applied to chirality and abelian extensions. The square and granny knots, for example, can be distinguished by quandle colorings, so that a trefoil and its mirror…