Related papers: A novel connection between integral binary quadrat…
In this paper, we give a formula for the proper class number of a binary quadratic polynomial assuming that the conductor ideal is sufficiently divisible at dyadic places. This allows us to study the growth of the proper class numbers of…
An invariant of knots is constructed from an integral for geometric braids due to Kohno and Kontsevich. It takes values in a quotient by a certain ideal of the algebra generated by chord diagrams over the circle.
We give a cabling formula for the Links--Gould polynomial of knots colored with a $4n$-dimensional irreducible representation of $\mathrm{U}^H_q\mathfrak{sl}(2|1)$ and identify them with the $V_n$-polynomial of knots for $n=2$. Using the…
In this paper we discuss the applications of knotoids to modelling knots in open curves and produce new knotoid invariants. We show how invariants of knotoids generally give rise to well-behaved measures of how much an open curve is…
We produce combinatorial formulas for invariants of smooth embeddings of $(2\ell-1)$-spheres into $\mathbb{R}^{3\ell}$ for $\ell\geq 2$. Furthermore, we obtain such a formula for the Haefliger invariant, which classifies smooth knots…
We show how the signed evaluations of link polynomials can be used to calculate unknotting numbers. We use the Jones-Rong value of the Brandt-Lickorish-Millett-Ho polynomial Q to calculate the unknotting numbers of 8_{16}, 9_{49} and 6…
Let $K_n$ be a complete graph with $n$ vertices. An embedding of $K_n$ in $S^3$ is called a spatial $K_n$-graph. Knots in a spatial $K_n$-graph corresponding to simple cycles of $K_n$ are said to be constituent knots. We consider the case…
We present a conjecture for the power-law exponent in the asymptotic number of types of plane curves as the number of self-intersections goes to infinity. In view of the description of prime alternating links as flype equivalence classes of…
In this paper we study the theory of knotoids and braidoids and the theory of pseudo knotoids and pseudo braidoids on the torus T. In particular, we introduce the notion of {\it mixed knotoids} in $S^2$, that generalize the notion of mixed…
In this paper, we construct quantum invariants for knotoid diagrams in $\mathbb{R}^2$. The diagrams are arranged with respect to a given direction in the plane ({\it Morse knotoids}). A Morse knotoid diagram can be decomposed into basic…
We partially determine grid homology (combinatorial knot Floer homology) of diagonal knots, which are conjectured to be equivalent to positive braid knots, by exploiting nice grid diagrams. Its next-to-top term detects the number of prime…
We show that 3-braid links with given (non-zero) Alexander or Jones polynomial are finitely many, and can be effectively determined. We classify among closed 3-braids strongly quasipositive and fibered ones, and show that 3-braid links have…
We introduce new obstructions to topological knot concordance. These are obtained from amenable groups in Strebel's class, possibly with torsion, using a recently suggested $L^2$-theoretic method due to Orr and the author. Concerning…
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…
Given any oriented link diagram, two types of new knot invariants are constructed. They satisfy some generalized skein relations. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations of those…
We construct an infinite tower of covering spaces over the configuration space of $n-1$ distinct non-zero points in the complex plane. This results in an action of the braid group $\mathbb{B}_n$ on the set of $n$-adic integers…
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…
We introduce a special class of knots, called global knots, in F^2 x R and we construct new isotopy invariants, called T-invariants, for global knots. Some T-invariants are of finite type but they cannot be extracted from the generalized…
This short survey, which was written to accompany a minicourse at the BIRS conference "Topology in dimension 4.5", concerns invariants of knotted $2$-spheres in $S^4$, also known as $2$-knots. It covers invariants extracted from the…
We prove that if the order of the first homology of the 2-fold branched cover of a knot K in the 3-sphere is given by pm where p is a prime congruent to 3 mod 4 and gcd(p,m) =1, then K is of infinite order in the knot concordance group.…