An Exact Sequence of Braid Groups.
An expansion of the Jones representation of genus 2 and the Torelli group.
An extension of the Loop theorem and resolutions of generalized 3-manifolds with 0-dimensional singular set.
An implementation of the Bestvina-Handel algorithm for surface homeomorphisms.
An indecomposable -complex. II.
An infinite torus braid yields a categorified Jones-Wenzl projector
A sequence of Temperley-Lieb algebra elements corresponding to torus braids with growing twisting numbers converges to the Jones-Wenzl projector. We show that a sequence of categorification complexes of these braids also has a limit which may serve as a categorification of the Jones-Wenzl projector.
An infinite-dimensional torsion-free FP...group.
An introduction to the abelian Reidemeister torsion of three-dimensional manifolds
These notes accompany some lectures given at the autumn school “Tresses in Pau” in October 2009. The abelian Reidemeister torsion for -manifolds, and its refinements by Turaev, are introduced. Some applications, including relations between the Reidemeister torsion and other classical invariants, are surveyed.
An invariant of link cobordisms from Khovanov homology.
An invariant of tangle cobordisms via subquotients of arc rings
We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).
An irreducible Heegaard diagram of the real projective 3-space .
An obstruction to sliceness via contact geometry and "classical" gauge theory.
An operator invariant for handlebody-knots
A handlebody-knot is a handlebody embedded in the 3-sphere. We improve Luo's result about markings on a surface, and show that an IH-move is sufficient to investigate handlebody-knots with spatial trivalent graphs without cut-edges. We also give fundamental moves with a height function for handlebody-tangles, which helps us to define operator invariants for handlebody-knots. By using the fundamental moves, we give an operator invariant.
An upper estimation for the eigenfrequencies of vibrating Liapunoff bodies (first boundary value problem).
Analyticity of the free energy of a closed 3-manifold.
Andreev’s Theorem on hyperbolic polyhedra
In 1970, E.M.Andreev published a classification of all three-dimensional compact hyperbolic polyhedra (other than tetrahedra) having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, , Andreev’s Theorem provides five classes of linear inequalities, depending on , for the dihedral angles, which are necessary and sufficient conditions for the existence of a hyperbolic polyhedron realizing with the assigned dihedral angles. Andreev’s Theorem also shows that the resulting...
Applications of topology to DNA
The following is an expository article meant to give a simplified introduction to applications of topology to DNA.
Approximating homotopy equivalences of surfaces by homeomorphisms
Approximation L2-invariants by Their Finite-Dimensional Analogues.
Arc presentations of knots and links
s paper presents some examples and a survey of results concerning a new way of presenting knots and links, together with the corresponding link invariant. More detailed accounts are given in [Cr, C-N, Nu1, Nu2, Nu3].