Related papers: The rational Witt class and the unknotting number …
Neuwirth asked if any non-trivial knot in the 3-sphere can be embedded in a closed surface so that the complement of the surface is a connected essential surface for the knot complement. In this paper, we examine some variations on this…
Let $M_n$ be the topological moduli space of all parallel n-cables of long framed oriented knots in 3-space. We construct in a combinatorial way for each natural number $n>1$ a 1-cocycle $R_n$ which represents a non trivial class in…
This thesis develops some general calculational techniques for finding the orders of knots in the topological concordance group C. The techniques currently available in the literature are either too theoretical, applying to only a small…
In this article we discuss applications of neural networks to recognising knots and, in particular, to the unknotting problem. One of motivations for this study is to understand how neural networks work on the example of a problem for which…
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…
The crosscap number of a knot is an invariant describing the non-orientable surface of smallest genus that the knot bounds. Unlike knot genus (its orientable counterpart), crosscap numbers are difficult to compute and no general algorithm…
For any knot with genus one and unknotting number one, other than the figure-eight knot, we prove that there is exactly one way to unknot it by means of a crossing change. In the case of the figure-eight knot, we prove that there are…
We prove that if an alternating 3-braid knot has unknotting number one, then there must exist an unknotting crossing in any alternating diagram of it, and we enumerate such knots. The argument combines the obstruction to unknotting number…
This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the…
We introduce natural language processing into the study of knot theory, as made natural by the braid word representation of knots. We study the UNKNOT problem of determining whether or not a given knot is the unknot. After describing an…
We study the properties of glued knots, a sub-class of real rational knots, that can be constructed by gluing ellipses. We define an invariant called the gluing degree and relate it to various classical properties of knots and classify all…
The virtual unknotting number of a virtual knot is the minimal number of crossing changes that makes the virtual knot to be the unknot, which is defined only for virtual knots virtually homotopic to the unknot. We focus on the virtual knot…
Let $u(K)$ and $g(K)$ denote the unknotting number and the genus of a knot $K$, respectively. For a 3-braid knot $K$, we show that $u(K)\le g(K)$ holds, and that if $u(K)=g(K)$ then $K$ is either a 2-braid knot, a connected sum of two…
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 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…
Given a knot in the three-sphere, is it possible to unknot it by performing a single twist, and if so, what are the possible linking numbers of such a twist? We develop obstructions to unknotting using a twist of a specified linking number.…
A knot in $S^3$ is rationally slice if it bounds a disk in a rational homology ball. We give an infinite family of rationally slice knots that are linearly independent in the knot concordance group. In particular, our examples are all…
A powerful way to study groups is via their actions on suitable spaces. Classifying spaces for families of subgroups are a type of these spaces, obtained by imposing some strict conditions on the fixed-point sets. We show how in the…
We study the group of rational concordance classes of codimension two knots in rational homology spheres. We give a full calculation of its algebraic theory by developing a complete set of new invariants. For computation, we relate these…
We discuss an "extrinsic" property of knots in a 3-subspace of the 3-sphere $S^3$ to characterize how the subspace is embedded in $S^3$. Specifically, we show that every knot in a subspace of the 3-sphere is transient if and only if the…