Related papers: The search for exotic knot traces

In this paper, we study a geometric/topological measure of knots and links called the nullification number. The nullification of knots/links is believed to be biologically relevant. For example, in DNA topology, one can intuitively regard…

We construct two complete invariants of oriented classical knots in space. The value of each invariant on any knot is a set, infinite for the first invariant and finite for the second. The finite set is computed algorithmically from any…

Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the…

We use classical techniques to answer some questions raised by Daniele Celoria about almost-concordance of knots in arbitrary closed $3$-manifolds. We first prove that, given $Y^3 \neq S^3$, for any non-trivial element $g\in \pi_1(Y)$ there…

We introduce a method to detect exotic surfaces without explicitly using a smooth 4-manifold invariant or an invariant of a 4-manifold-surface pair in the construction. Our main tools are two versions of families (Seiberg-Witten)…

It is well known that there exist knots with Seifert surfaces of arbitrarily high genus. In this paper, we show the existence of infinitely many knot exteriors where each of which has longitudinal essential surfaces of any positive genus…

We provide an approach to study exotic phenomena in relatively small 4-manifolds that captures many different exotic behaviors under one umbrella. These phenomena include exotic smooth structures on 4-manifolds with $b_2=1$, examples of…

A {\it stuck knot} is a knot diagram containing designated crossings, called {\it stuck crossings}, whose incident strands are required to remain locally non-separable. These rigidity constraints restrict the allowable ambient isotopies and…

This is an expository article of our work on analogies between knot theory and algebraic number theory. We shall discuss foundational analogies between knots and primes, 3-manifolds and number rings mainly from the group-theoretic point of…

The backbone of most proteins forms an open curve. To study their entanglement, a common strategy consists in searching for the presence of knots in their backbones using topological invariants. However, this approach requires to close the…

Query answering routinely employs knowledge graphs to assist the user in the search process. Given a knowledge graph that represents entities and relationships among them, one aims at complementing the search with intuitive but effective…

We study the formation of knots on a macroscopic ball-chain, which is shaken on a horizontal plate at 12 times the acceleration of gravity. We find that above a certain critical length, the knotting probability is independent of chain…

We propose a friend recommendation system (an application of link prediction) using edge embeddings on social networks. Most real-world social networks are multi-graphs, where different kinds of relationships (e.g. chat, friendship) are…

These notes present two normal surface theory algorithms to detect the unknot and use the split-link algorithm to prove that the figure-eight knot is knotted.

We report on the geometry and mechanics of knotted stiff strings. We discuss both closed and open knots. Our two main results are: (i) Their equilibrium energy as well as the equilibrium tension for open knots depend on the type of knot as…

We describe a construction procedure of infinite sets of $2$-links in closed simply connected 4-manifolds that are topologically isotopic, smoothly inequivalent and componentwise topologically unknotted. These 2-links are the first examples…

Distinct knots K, K' can sometimes share a common p/q-framed Dehn surgery. A folk conjecture held that for a fixed pair of knots, this can occur for at most one value of p/q. We disprove this conjecture by constructing pairs of distinct…

It is known that every exotic smooth structure on a simply connected closed 4-manifold is determined by a codimention zero compact contractible Stein submanifold and an involution on its boundary. Such a pair is called a cork. In this…

We show that there exists an infinite family of pairwise non-isotopic Legendrian knots in the standard contact 3-sphere whose Stein traces are equivalent. This is the first example of such phenomenon. Different constructions are developed…

We describe a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors. Recently this technique…