Related papers: On 2-bridge knots with differing smooth and topolo…

We develop a topological model of knots and links arising from a single (or multiple processive) round(s) of recombination starting with an unknot, unlink, or (2,m)-torus knot or link substrate. We show that all knotted or linked products…

Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which,…

We show that a two-bridge ribbon knot $K(m^2 , m k \pm 1)$ with $m > k >0$ and $(m,k)=1$ admits a symmetric union presentation with partial knot which is a two-bridge knot $K(m,k)$. Similar descriptions for all the other two-bridge ribbon…

For some families of two-bridge knots, including double-twist knots with genus at least four, we determine precisely the set of integers $n>1$ such that the fundamental group of the $n$-fold cyclic branched cover of the 3-sphere along these…

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…

Let $p$ be a prime number and let $K$ be a genus one two-bridge knot. In the spirit of arithmetic topology, we observe that if $p$ divides the size of the 1st homology group of some odd-th cyclic branched cover of the knot $K$, then its…

Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long exact sequences in homology and further algebraic structure which is then used to…

We show that a 2-knot group discovered in the course of a census of 4-manifolds with small triangulations is an HNN extension with finite base and proper associated subgroups, and has the smallest base among such knot groups.

Simple closed curves in the plane can be mapped to nontrivial knots under the action of origami foldings that allow the paper to self-intersect. We show all tame knot types may be produced in this manner, motivating the development of a new…

A knot is said to be slice if it bounds a smooth properly embedded disk in the 4-ball. We demonstrate that the Conway knot, 11n34 in the Rolfsen tables, is not slice. This completes the classification of slice knots under 13 crossings, and…

We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots…

Ng constructed an invariant of knots in ${\mathbb{R}}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${\mathbb{R}}^4$ using marked graph diagrams.

Suppose $K$ is a knot in $S^3$ with bridge number $n$ and bridge distance greater than $2n$. We show that there are at most ${2n\choose n}$ distinct minimal genus Heegaard splittings of $S^3\setminus\eta(K)$. These splittings can be divided…

We determine a simple condition on a particular state graph of an alternating knot or link diagram that characterizes when the unoriented genus and crosscap number coincide, extending work of Adams and Kindred. Building on this same work…

Although there are infinitely many knots with superbridge index n for every even integer n>2, there are only finitely many knots with superbridge index 3.

We prove that if an alternating knot has unknotting number one, then there exists an unknotting crossing in any alternating diagram. This is done by showing that the obstruction to unknotting number one developed by Greene in his work on…

We analyze the orbifolds that can be obtained as quotients of hyperbolic 3-manifolds admitting a Heegaard splitting of genus two by their orientation preserving isometry groups. The genus two hyperbolic 3-manifolds are exactly the…

Generalizing Howie and Greene's characterization of alternating knots, we give a topological characterization of almost alternating knots.

The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. We determine the $\Delta$-unknotting numbers for two-bridge knots of type $C(2\beta_1, 2\beta_2,…

We determine the set of all genus g bridge numbers of many iterated torus knots, listing these numbers in a sequence called the bridge spectrum. In addition, we prove a structural lemma about the decomposition of a strongly irreducible…