On 3-manifold invariants arising from finite-dimensional Hopf algebras.
On a secondary invariant of the hyperelliptic mapping class group
We discuss relations among several invariants of 3-manifolds including Meyer's function, the η-invariant, the von Neumann ρ-invariant and the Casson invariant from the viewpoint of the mapping class group of a surface.
On a simple invariant of Turaev-Viro type
On a theorem of Kontsevich.
On bilinear biquandles
We define a type of biquandle which is a generalization of symplectic quandles. We use the extra structure of these bilinear biquandles to define new knot and link invariants and give some examples.
On Chern-Simons invariants of geometric 3-manifolds.
On iterated torus knots and transversal knots.
On Khovanov's categorification of the Jones polynomial.
On knot Floer homology and cabling.
On numerical invariants for knots in the solid torus
We define some new numerical invariants for knots with zero winding number in the solid torus. These invariants explore some geometric features of knots embedded in the solid torus. We characterize these invariants and interpret them on the level of the knot projection. We also find some relations among some of these invariants. Moreover, we give lower bounds for some of these invariants using Vassiliev invariants of type one. We connect our invariants to the bridge number in the solid torus. We...
On small geometric invariants of 3-manifolds.
On some ternary operations in knot theory
We introduce a way to color the regions of a classical knot diagram using ternary operations, so that the number of colorings is a knot invariant. By choosing appropriate substitutions in the algebras that we assign to diagrams, we obtain the relations from the knot group, and from the core group. Using the ternary operator approach, we generalize the Dehn presentation of the knot group to extra loops, and a similar presentation for the core group to the variety of Moufang loops.
On stability of Alexander polynomials of knots and links (survey)
We study distribution of the zeros of the Alexander polynomials of knots and links in S³. After a brief introduction of various stabilities of multivariate polynomials, we present recent results on stable Alexander polynomials.
On the colored Jones polynomials of ribbon links, boundary links and Brunnian links
Habiro gave principal ideals of in which certain linear combinations of the colored Jones polynomials of algebraically-split links take values. The author proved that the same linear combinations for ribbon links, boundary links and Brunnian links are contained in smaller ideals of generated by several elements. In this paper, we prove that these ideals also are principal, each generated by a product of cyclotomic polynomials.
On the complexity of graph-manifolds.
On the cut number of a 3-manifold.
On the dynamics of (left) orderable groups
We develop dynamical methods for studying left-orderable groups as well as the spaces of orderings associated to them. We give new and elementary proofs of theorems by Linnell (if a left-orderable group has infinitely many orderings, then it has uncountably many) and McCleary (the space of orderings of the free group is a Cantor set). We show that this last result also holds for countable torsion-free nilpotent groups which are not rank-one Abelian. Finally, we apply our methods to the case of braid...
On the first two Vassiliev invariants.
On the Floer homology of plumbed three-manifolds.
On the kauffman bracket skein algebra of parallelized surfaces