Related papers: Survey on recent invariants on classical knot theo…

This review is an extended version of the Seoul ICM 2014 proceedings.It is a short overview of the "topological recursion", a relation appearing in the asymptotic expansion of many integrable systems and in enumerative problems. We recall…

In 1993, just about a century after the epoch of Classical Invariant Theory and almost 30 years after Mumford's seminal book on Geometric Invariant Theory, Bernd Sturmfels approached the subject from a new, algorithmic perspective in his…

This monograph derives direct and concrete relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses that arise naturally in the study…

We construct knot invariants from solutions to the Yang--Baxter equation associated to appropriately generalized left/right Yetter--Drinfel'd modules over a braided Hopf algebra with an automorphism. When applied to Nichols algebras, our…

In 1999, Khovanov showed that a link invariant known as the Jones polynomial is the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their…

In this paper, we discuss a proof of the isotopy invariance of a parametrized Khovanov link homology including categorifications of the Jones polynomial and the Kauffman bracket polynomial though it is a known fact. In order to present a…

Chern-Simons theory in the 1/N expansion has been conjectured to be equivalent to a topological string theory. This conjecture predicts a remarkable relationship between knot invariants and Gromov-Witten theory. We review some basic aspects…

The discovery of polynomial invariants of knots and links, ignited by V. F. R. Jones, leads to the formulation of polynomial invariants of spatial graphs. The Yamada polynomial, one of such invariants, is frequently utilized for practical…

Geometric interpretations of some virtual knot invariants are given in terms of invariants of links in $\mathbb{S}^3$. Alexander polynomials of almost classical knots are shown to be specializations of the multi-variable Alexander…

Knot Floer homology is an invariant for knots discovered by the authors and, independently, Jacob Rasmussen. The discovery of this invariant grew naturally out of studying how a certain three-manifold invariant, Heegaard Floer homology,…

Notes based on lessons given at {\sc Escuela " Fico Gonz\'alez Acu\~na" de Nudos y 3-variedades}, M\'erida Yucat\'an, M\'exico, 7--10 (2015) and {\sc Encuentro de nudos, trenzas y \'algebras}, Oaxaca--M\'exico, 3--10 October (2018).

We study random knots and links in R^3 using the Petaluma model, which is based on the petal projections developed by Adams et al. (2012). In this model we obtain a formula for the distribution of the linking number of a random…

Kronheimer and Mrowka introduced a new knot invariant, called $s^\sharp$, which is a gauge theoretic analogue of Rasmussen's $s$ invariant. In this article, we compute Kronheimer and Mrowka's invariant for some classes of knots, including…

Closed geodesics associated with indefinite binary quadratic forms, or equivalently with real quadratic irrationals, have long been studied as geometric $\mathrm{SL}_2(\mathbb{Z})$-invariants. Building on the Birman-Williams approach to…

Gram determinants earned traction among knot theorists after E. Witten's presumption about the existence of a 3-manifold invariant connected to the Jones polynomial. Triggered by the creation of such an invariant by N. Reshetikhin and V.…

We develop an invariant of knots that depends on a complex parameter t, describing a left ideal in the noncommutative torus. When the parameter is set equal to -1 we recover the A-polynomial of the knot. We relate the invariant to the…

Knots and links in 3-manifolds are studied by applying intersection invariants to singular concordances. The resulting link invariants generalize the Arf invariant, the mod 2 Sato-Levine invariants, and Milnor's triple linking numbers.…

Link homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We introduce a new equivalence relation on spatial graphs called component homotopy, which reduces to link homotopy in the…

A classical knot is described by a one-stroke trajectory with entanglements of a string. The replica method appears as a powerful tool in statistical mechanics for a polymer or self-avoiding walk. We consider this replica N to 0 limit in…

Consider a continuous flow in $\mathbb{R}^3$ or any orientable $3$-manifold. Let $(Q_1, Q_0)$ be an index pair in the sense of Conley and consider the region $N := \overline{Q_1 - Q_0}$. (An example of this is a compact $3$-manifold $N$…