The Bloch invariant as a characteristic class in .
The colored Jones function is -holonomic.
The colored Jones polynomials of the figure-eight knot and the volumes of three-manifolds obtained by Dehn surgeries
I describe how the colored Jones polynomials of the figure-eight knot determine the volumes of the three-manifolds obtained by Dehn surgeries along it, according to my joint work with Y. Yokota.
The configuration space integral for links in .
The co-rank conjecture for 3-manifold groups.
The coribbon structures of some finite dimensional braided Hopf algebras generated by 2×2-matrix coalgebras
We determine the coribbon structures of some finite dimensional braided Hopf algebras generated by 2×2-matrix coalgebras constructed by S. Suzuki. As a consequence, we see that such a Hopf algebra has a coribbon structure if and only if it is of Kac-Paljutkin type.
The determinant of oriented rotants
We study the determinant of pairs of rotants of Anstee, Przytycki and Rolfsen. We consider various notions of rotant orientations.
The diagrammatic Soergel category and -foams, for .
The disjoint curve property.
The equation [B,(A-1)(A,B)] = 0 and virtual knots and links
Let A, B be invertible, non-commuting elements of a ring R. Suppose that A-1 is also invertible and that the equation [B,(A-1)(A,B)] = 0 called the fundamental equation is satisfied. Then this defines a representation of the algebra ℱ = A, B | [B,(A-1)(A,B)] = 0. An invariant R-module can then be defined for any diagram of a (virtual) knot or link. This halves the number of previously known relations and allows us to give a complete solution in the case when R is the quaternions.
The Fukumoto-Furuta and the Ozsváth-Szabó invariants for spherical 3-manifolds
We show that the Fukumoto-Furuta invariant for a rational homology 3-sphere M, which coincides with the Neumann-Siebenmann invariant for a Seifert rational homology 3-sphere, is the same as the Ozsváth-Szabó's correction term derived from the Heegaard Floer homology theory if M is a spherical 3-manifold.
The geometric genus of hypersurface singularities
Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non-degenerate hypersurface singularities.
The Homflypt skein module of a connected sum of 3-manifolds.
The Kauffman bracket skein module of a twist knot exterior.
The Kontsevich integral and quantized Lie superalgebras.
The Kontsevich integral and re-normalized link invariants arising from Lie superalgebras
We show that the coefficients of the re-normalized link invariants of [3] are Vassiliev invariants which give rise to a canonical family of weight systems.
The number of independent Vassiliev invariants in the Homfly and Kauffman polynomials.
The orbits of the Hurwitz action of the braid groups on the standard generators
The Hurwitz action of the n-braid group Bₙ on the n-fold direct product of the m-braid group is studied. We show that the orbit of any n- tuple of the n standard generators of consists of the (n-1)th powers of n+1 elements.
The proof of Birman's conjecture on singular braid monoids.
The reduction of quantum invariants of 4-thickenings
We study the sensibility of an invariant of 2-dimensional CW complexes in the case when it comes as a reduction (through a change of ring) of a modular invariant of 4-dimensional thickenings of such complexes: it is shown that if the Euler characteristic of the 2-complex is greater than or equal to 1, its invariant depends only on homology. To see what is happening when the Euler characteristic is smaller than 1, we use ideas of Kerler and construct, from any tortile category, an invariant of 4-thickenings...