Algebraic linking numbers of knots in 3-manifolds.
All integral slopes can be Seifert fibered slopes for hyperbolic knots.
All Knot Groups Are Metric.
All knots are algebraic.
All two dimensions links are null homotopic.
An Algebraic Classification of some Knots of Codimension Two.
An algorithm of finding planar surfaces in three-manifolds.
An algorithm to detect laminar 3-manifolds.
An approach to automorphisms of free groups and braids via transvections.
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 invariant of link cobordisms from Khovanov homology.
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.
Analyticity of the free energy of a closed 3-manifold.
Applications of topology to DNA
The following is an expository article meant to give a simplified introduction to applications of topology to DNA.
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].
Arc-presentations of links: Monotonic simplification
In the early 90's J. Birman and W. Menasco worked out a nice technique for studying links presented in the form of a closed braid. The technique is based on certain foliated surfaces and uses tricks similar to those that were introduced earlier by D. Bennequin. A few years later P. Cromwell adapted Birman-Menasco's method for studying so-called arc-presentations of links and established some of their basic properties. Here we further develop that technique and the theory of arc-presentations, and...
Around the Borromean link.
This is a survey of some consequences of the fact that the fundamental group of the orbifold with singular set the Borromean link and isotropy cyclic of order 4 is a universal kleinian group.