Related papers: On transversally simple knots

We introduce a 4-dimensional analogue of the rational Seifert genus of a knot $K\subset Y$, which we call the rational slice genus, that measures the complexity of a homology class in $H_2(Y\times [0,1],K;\mathbb{Q})$. Our main theorem is a…

The stable 4-genus of a knot K in 3-space is the limiting value of g_4(nK)/n, where g_4 denotes the 4-genus and n goes to infinity. This induces a seminorm on CQ, the concordance group tensored with the rational numbers. Basic properties of…

Topological entanglements are abundant, and often detrimental, in polymeric systems in biology and materials science. Here we theoretically investigate the topological simplification of knots by diffusing slip-links (SLs), which may…

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 k is called ``strongly (n-1)-trivial.'' if there exists a projection of k, such that one can choose n crossings of the projection with the property that making the crossing changes corresponding to any of the $2^{n}-1$ nontrivial…

Let K be a nontrivial knot in the 3-sphere with the exterior E(K), and u in G(K), the fundamental group of E(K), a slope element represented by an essential simple closed curve on the boundary of E(K). Since the normal closure of u in G(K)…

Given knots K and J, one can ask whether a single smoothing of a crossing in a diagram for K can convert it into a diagram for J. As an interesting example, Zekovic discovered that the torus knot T(2,5) can be converted into T(2,-5) with a…

We consider the space of all smooth knots in the 3-sphere isotopic to a given knot, with the aim of finding a small subspace onto which this large space deformation retracts. For torus knots and many hyperbolic knots we show the subspace…

The non-orientable 4-genus of a knot $K$ in $S^{3}$, denoted $\gamma_4(K)$, measures the minimum genus of a non-orientable surface in $B^{4}$ bounded by $K$. We compute bounds for the non-orientable 4-genus of knots $T_{5, q}$ and $T_{6,…

The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of…

The study of knots and links from a probabilistic viewpoint provides insight into the behavior of "typical" knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of…

Let $\mathcal {M}$ be the space of all, including singular, long knots in 3-space and for which a fixed projection into the plane is an immersion. Let $cl(\Sigma^{(1)}_{iness})$ be the closure of the union of all singular knots in $\mathcal…

We present an enhanced prime decomposition theorem for knots that gives the isotopy classes of composite knots that can be constructed from a given list of prime factors (allowing for the mirroring and orientation reversing for each…

It is proved that the Wedderburn Theorem on finite division rings implies that all knots and links in the smooth 4-dimensional manifolds are trivial.

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…

In this paper we show that the non-alternating torus knots are homologically thick, i.e. that their Khovanov homology occupies at least three diagonals. Furthermore, we show that we can reduce the number of full twists of the torus knot…

A conjecture of Shumakovitch states that every nontrivial knot has 2-torsion in its Khovanov homology. We show that if a knot $K$ has no 2-torsion in its Khovanov homology, then the rank of its reduced Khovanov homology is minimal among all…

We prove that each overtwisted contact structure has knot types that are represented by infinitely many distinct transverse knots all with the same self-linking number. In some cases, we can even classify all such knots. We also show…

Using convex surfaces and Kanda's classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on $T^3$. Some of the knot types considered in this article provide new examples of…

We study simple, knotted and linked torus windings that are made of tubes of finite thickness. Knots which have the shortest rope length are often denoted ideal structures. Conventionally, the ideal structure are found by rope shortening…