Related papers: Learning knot invariants across dimensions
There is a higher dimensional analogue of the perturbative Chern-Simons theory in the sense that a similar perturbative series as in 3-dimension, which is computed via configuration space integral, yields an invariant of higher dimensional…
We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to…
We define invariants for a framed link equipped with a SL2 local system in its complement and additional combinatorial data based on the theory of representations of stated skein algebras at roots of unity of punctured bigons and the…
We construct a new family of knot concordance invariants $\theta^{(q)}(K)$, where $q$ is a prime number. Our invariants are obtained from the equivariant Seiberg-Witten-Floer cohomology, constructed by the author and Hekmati, applied to the…
In this paper we compute the signature for a family of knots $W(k,n)$, the weaving knots of type $(k,n)$. By work of E.~S.~Lee the signature calculation implies a vanishing theorem for the Khovanov homology of weaving knots. Specializing to…
We describe an invariant of links in the three-sphere which is closely related to Khovanov's Jones polynomial homology. Our construction replaces the symmetric algebra appearing in Khovanov's definition with an exterior algebra. The two…
We define an invariant of transverse links in the standard contact 3-sphere as a distinguished element of the Khovanov homology of the link. The quantum grading of this invariant is the self-linking number of the link. For knots, this gives…
Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. For symmetric diagrams we develop a two-variable refinement $W_D(s,t)$ of the Jones polynomial that…
This work is dedicated to the consideration of the construction of a representation of braid group generators from vertex models with $N$-states, which provides a great way to study the knot invariant. An algebraic formula is proposed for…
Besides offering a friendly introduction to knot homologies and quantum curves, the goal of these lectures is to review some of the concrete predictions that follow from the physical interpretation of knot homologies. In particular, this…
Quantum knot invariants (like colored HOMFLY-PT or Kauffman polynomials) are a distinguished class of non-perturbative topological invariants. Any known way to construct them (via Chern-Simons theory or quantum R-matrix) starts with a…
We give a combinatorial treatment of transverse homology, a new invariant of transverse knots that is an extension of knot contact homology. The theory comes in several flavors, including one that is an invariant of topological knots and…
In this report, I will start by first giving a brief introduction on knots to build some intuition before beginning the more rigorous review in the Literature Review section. There, I will define knot equivalence, the Jones polynomial…
We study inequalities between integer-valued knot invariants arising from classical knot theory, four-dimensional topology, knot homologies, and knot polynomials. We present a directed graph consisting of 48 inequalities between 33 knot…
In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation.…
We reveal an intimate connection between the quantum knot invariant for torus knot T(s,t) and the character of the minimal model M(s,t), where s and t are relatively prime integers. We show that Kashaev's invariant, i.e., the N-colored…
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…
The colored Jones polynomial $J_{K,N}$ is an important quantum knot invariant in low-dimensional topology. In his seminal paper on quantum modular forms, Zagier predicted the behavior of $J_{K,0}(e^{2 \pi i x})$ under the action of…
We introduce new skein invariants of links based on a procedure where we first apply the skein relation only to crossings of distinct components, so as to produce collections of unlinked knots. We then evaluate the resulting knots using a…
We construct geometrically two universal link invariants: universal ADO invariant and universal Jones invariant, as limits of invariants given by graded intersections in configuration spaces. More specifically, for a fixed level $\mathscr…