Related papers: A lower bound on critical points of the electric p…
Physical knot theory, where knots are treated like physical objects, is important to many fields. One natural problem is to give a knot a uniform charge, and analyze the resulting electric field and electric potential. There have been some…
Consider a knot $K$ in $S^3$ with uniformly distributed electric charge. Whilst solutions to the Laplace equation in terms of Dirichlet integrals are readily available, it is still of theoretical and physical interest to understand the…
The ropelength of a knot is the quotient of its length by its thickness. We consider a family of energy functions for knots, depending on a power p, which approach ropelength as p increases. We describe a numerically computed trefoil knot…
The ropelength of a knot is the minimum contour length of a tube of unit radius that traces out the knot in three dimensional space without self-overlap, colloquially the minimum amount of rope needed to tie a given knot. Theoretical upper…
The transient number of a knot K, denoted tr(K), is the minimal number of simple arcs that have to be attached to K, in order that K can be homotoped to a trivial knot in a regular neighborhood of the union of K and the arcs. We give a…
For a knot or link K, let L(K) be the ropelength of K and Cr(K) be the crossing number of K. In this paper, we show that there exists a constant a>0 such that L(K) is bounded above by a Cr(K) ln^5 (Cr(K)) for any knot K. This result shows…
The warping sum $e(K)$ of a knot $K$ is the minimal value of the sum of the warping degrees of a minimal diagram of $K$ with both orientations. In this paper, knots $K$ with $e(K) \le 3$ are characterized, and some knots $K$ with $e(K)=4$…
The theory of tunnel number 1 knots detailed in our previous paper, The tree of knot tunnels, provides a non-negative integer invariant called the depth of the tunnel. We give various results related to the depth invariant. Noting that it…
The main open problem in geometric knot theory is to provide a tabulation of knots based on an energy criterion, with the goal of presenting this tabulation in terms of global energy minimisers within isotopy classes, often referred to as…
The ribbon number $r(K)$ of a ribbon knot $K \subset S^3$ is the minimal number of ribbon intersections contained in any ribbon disk bounded by $K$. We find new lower bounds for $r(K)$ using $\det(K)$ and $\Delta_K(t)$, and we prove that…
The probability of a random polygon (or a ring polymer) having a knot type $K$ should depend on the complexity of the knot $K$. Through computer simulation using knot invariants, we show that the knotting probability decreases exponentially…
The ropelength problem asks for the minimum-length configuration of a knotted diameter-one tube embedded in Euclidean three-space. The core curve of such a tube is called a tight knot, and its length is a knot invariant measuring…
We introduce two numerical invariants, the waist and the trunk of knots. The waist of a closed incompressible surface in the complement of a knot is defined as the minimal intersection number of all compressing disks for the surface in the…
A long standing open conjecture states that if a link $\mathcal{K}$ is alternating, then its ropelength $L(\mathcal{K})$ is at least of the order $O(Cr(\mathcal{K}))$. A recent result shows that the maximum braid index of a link bounds the…
We give an alternative proof of that a critical knot of a Morse-Bott function $f: S^3 \rightarrow \mathbb{R}$ is a graph knot where the critical set of $f$ is a link in $S^3$. Our proof inducts on the number of index-1 critical knots of…
We demonstrate the existence of stable knot solitons in the standard electroweak theory whose topological quantum number $\pi_3(S^2)$ is fixed by the Chern-Simon index of the $Z$ boson. The electroweak knots are made of the helical magnetic…
Knots are commonly found in molecular chains such as DNA and proteins, and they have been considered to be useful models for structural analysis of these molecules. One interested quantity is the minimum number of monomers necessary to…
Let $\mbox{Len}(K)$ be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for $\mbox{Len}(K)$ of a nontrivial knot $K$ in…
The ropelength of a knot is the quotient of its length and its thickness, the radius of the largest embedded normal tube around the knot. We prove existence and regularity for ropelength minimizers in any knot or link type; these are…
Knots have been considered to be useful models for simulating molecular chains such as DNA and proteins. One quantity that we are interested on molecular knots is the minimum number of monomers necessary to realize a knot. In this paper we…