Journal of Knot Theory and Its Ramifications

We show that for any n ≥ 4 there exists an equivalence functor from the category of n-fold connected simple coverings of B³ × [0, 1] branched over ribbon surface tangles up to certain local ribbon moves, and the cobordism category of orientable relative 4-dimensional 2-handlebody cobordisms up to 2-deformations. As a consequence, we obtain an equivalence theorem for simple coverings of S³ branched over links, which provides a complete solution to the long-standing Fox–Montesinos covering moves problem. This last result generalizes to coverings of any degree results by the second author and Apostolakis, concerning respectively the case of degree 3 and 4. We also provide an extension of the equivalence theorem to possibly non-simple coverings of S³ branched over embedded graphs. Then, we factor the functor above as , where is an equivalence functor to a universal braided category freely generated by a Hopf algebra object H. In this way, we get a complete algebraic description of the category . From this we derive an analogous description of the category of 2-framed relative 3-dimensional cobordisms, which resolves a problem posed by Kerler.

In the 1920's Artin defined the braid group, B_n, in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of this correspondence, the Jones and Alexander polynomials, two of the most important knot invariants, can be described completely using the braid group. There has been a recent growth of interest in other diagrammatic algebras, whose elements have a similar topological flavor to the braid group. These have wide ranging applications in areas including representation theory and quantum computation. We consider representations of the braid group when passed through another diagrammatic algebra, the planar rook algebra. By studying traces of these matrices, we recover both the Jones and Alexander polynomials.

We go along a knot diagram, and get a sequence of over- and under- crossing points. We will study which kinds of sequences are realized by diagrams of the trefoil knot. As an application, we will characterize the Shimizu warping polynomials for diagrams of the trefoil knot.