Related papers: Minimal hard surface-unlink and classical unlink d…
We consider a natural model of random knotting- choose a knot diagram at random from the finite set of diagrams with n crossings. We tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type, allowing us to…
We seek to connect ideas in the theory of bridge trisections with other well-studied facets of classical knotted surface theory. First, we show how the normal Euler number can be computed from a tri-plane diagram, and we use this to give a…
It is known that each knot has a semimeander diagram (i. e. a diagram composed of two smooth simple arcs), however the number of crossings in such a diagram can only be roughly estimated. In the present paper we provide a new estimate of…
A $3$-connected graph is minimally 3-connected if removal of any edge destroys 3-connectivity. We present an algorithm for constructing minimally 3-connected graphs based on the results in (Dawes, JCTB 40, 159-168, 1986) using two…
The list of knots with up to 10 crossings is commonly referred to as the Rolfsen Table. This paper presents a way to generate the Rolfsen table in a simple, clear, and reproducible manner. The methods we use are similar to those used by J.…
A knot is a circle piecewise-linearly embedded into the 3-sphere. The topology of a knot is intimately related to that of its exterior, which is the complement of an open regular neighborhood of the knot. Knots are typically encoded by…
We have developed a reinforcement learning agent that often finds a minimal sequence of unknotting crossing changes for a knot diagram with up to 200 crossings, hence giving an upper bound on the unknotting number. We have used this to…
Deformations of knots and links in ambient space can be studied combinatorially on their diagrams via local modifications called Reidemeister moves. While it is well-known that, in order to move between equivalent diagrams with Reidemeister…
We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the…
A classification of spanning surfaces for alternating links is provided up to genus, orientability, and a new invariant that we call aggregate slope. That is, given an alternating link, we determine all possible combinations of genus,…
We classify graphs that are 0, 1, or 2 edges short of being complete partite graphs with respect to intrinsic linking and intrinsic knotting. In addition, we classify intrinsic knotting of graphs on 8 vertices. For graphs in these families,…
We prove that any diagram of the unknot with c crossings may be reduced to the trivial diagram using at most (236 c)^{11} Reidemeister moves. Moreover, every diagram in this sequence has at most (7 c)^2 crossings. We also prove a similar…
We will strengthen the known upper and lower bounds on the delta-crossing number of knots in therms of the triple-crossing number. The latter bound turns out to be strong enough to obtain (unknown values of) triple-crossing numbers for a…
Twisted links are a generalization of classical links and correspond to stably equivalence classes of links in thickened surfaces. In this paper we introduce twisted intersection colorings of a diagram and construct two invariants of a…
An interesting problem in classical differential geometry is to find methods to prove that two surfaces defined by different charts actually coincide up to position in space. In a previous paper we proposed a method in this direction for…
Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that…
Our main result is a version of Birman's theorem about equivalence of plats, which does not involve stabilization, for the unlink. We introduce the pocket and flip moves, which modify a plat without changing its link type or bridge index.…
Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded…
We describe a new class of minimal link diagrams. This class includes certain alternating diagrams, the standard diagrams of all torus links, and numerous homogeneous diagrams whose minimality has not been proven before. Besides, we…
We present a practical algorithm to determine the minimal genus of non-orientable spanning surfaces for 2-bridge knots, called the crosscap numbers. We will exhibit a table of crosscap numbers of 2-bridge knots up to 12crossings (all 362 of…