### $(1,1)$-knots via the mapping class group of the twice punctured torus.

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We describe a combinatorial algorithm for constructing all orientable 3-manifolds with a given standard bidimensional spine by making use of the idea of bijoin (Bandieri and Gagliardi (1982), Graselli (1985)) over a suitable pseudosimplicial triangulation of the spine.

We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labeled by irreducible representations of ${U}_{q}\left(sl\left(2\right)\right)$. We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the corresponding theory is a categorification of the colored Jones polynomial, and provides a tool for efficient computations of the resulting colored invariant of knots and links. Our theory is...

In this note, we prove the existence of a tri-graded Khovanov-type bicomplex (Theorem 1.2). The graded Euler characteristic of the total complex associated with this bicomplex is the colored Jones polynomial of a link. The first grading of the bicomplex is a homological one derived from cabling of the link (i.e., replacing a strand of the link by several parallel strands); the second grading is related to the homological grading of ordinary Khovanov homology; finally, the third grading is preserved...

We express the signature of an alternating link in terms of some combinatorial characteristics of its diagram.

We investigate the Khovanov-Rozansky invariant of a certain tangle and its compositions. Surprisingly the complexes we encounter reduce to ones that are very simple. Furthermore, we discuss a "local" algorithm for computing Khovanov-Rozansky homology and compare our results with those for the "foam" version of sl₃-homology.

We formulate a conjectural formula for Khovanov's invariants of alternating knots in terms of the Jones polynomial and the signature of the knot.

A category $N$ generalizing Jaeger-Nomura algebra associated to a spin model is given. It is used to prove some equivalence among the four conditions by Jaeger-Nomura for spin models of index 2.

The present paper is a continuation of our previous paper [Topology 44 (2005), 747-767], where we extended the Burau representation to oriented tangles. We now study further properties of this construction.