Journal of Knot Theory and Its Ramifications

Knitted and woven textile structures are examples of doubly periodic structures in a thickened plane made out of intertwining strands of yarn. Factoring out the group of translation symmetries of such a structure gives rise to a link diagram in a thickened torus, as in [2]. Such a diagram on a standard torus in S³ is converted into a classical link by including two auxiliary components which form the cores of the complementary solid tori. The resulting link, called a kernel for the structure, is determined by a choice of generators u, v for the group of symmetries.

A normalized form of the multi-variable Alexander polynomial of a kernel is used to provide polynomial invariants of the original structure which are essentially independent of the choice of generators u and v. It gives immediate information about the existence of closed curves and other topological features in the original textile structure. Because of its natural algebraic properties under coverings we can recover the polynomial for kernels based on a proper subgroup from the polynomial derived from the full symmetry group of the structure. This enables two structures to be compared at similar scales, even when one has a much smaller minimal repeating cell than the other.

Examples of simple traditional structures are given, and their Alexander data polynomials are presented to illustrate the techniques and results.

We found a family of infinitely many hyperbolic knot manifolds each member of which has a strongly detected boundary slope with associated root of unity of order 4.

We classify fibred closed 3-braids. In particular, given a nontrivial closed 3-braid, either it is fibred, or it differs from a fibred link by adding a crossing. The proof uses Gabai's method of disk decomposition. The topmost term in the knot Floer homology of closed 3-braids is also computed.

In A Volumish Theorem for the Jones Polynomial, by O. Dasbach and X. S. Lin, it was shown that the sum of the absolute value of the second and penultimate coefficient of the Jones polynomial of an alternating knot is equal to the twist number of the knot. Here we give a new proof of this result using Khovanov homology. The proof is by induction on the number of crossings using the long exact sequence in Khovanov homology corresponding to the Kauffman bracket skein relation.

In view of the result of Kontsevich, now often called "the fundamental theorem of Vassiliev theory", identifying the graded dual of the associated graded vector space to the space of Vassiliev invariants filtered by degree with the linear span of chord diagrams modulo the "4T-relation" (and in the unframed case, originally considered by Vassiliev, the "1T-" or "isolated chord relation"), it is a problem of some interest to provide a basis for the space of chord diagrams modulo the 4T-relation.

We construct the basis for the vector space spanned by chord diagrams with n chords and m link components, modulo 4T relations for n ≤ 5.

A knot diagram has an associated looped interlacement graph, obtained from the intersection graph of the Gauss diagram by attaching loops to the vertices that correspond to negative crossings. This construction suggests an extension of the Kauffman bracket to an invariant of looped graphs, and an extension of Reidemeister equivalence to an equivalence relation on looped graphs. The graph bracket polynomial can be defined recursively using the same pivot and local complementation operations used to define the interlace polynomial, and it gives rise to a graph Jones polynomial V_G(t) that is invariant under the graph Reidemeister moves.

It is well known that there is a one-to-one correspondence between signed plane graphs and link diagrams via the medial construction. The component number of the corresponding link diagram is however independent of the signs of the plane graph. Determining this number may be one of the first problems in studying links by using graphs. Some works in this aspect have been done. In this paper, we investigate the component number of links corresponding to lattices. Firstly we provide some general results on component number of links. Then, via these results, we proceed to determine the component number of links corresponding to lattices with free or periodic boundary conditions and periodic lattices with one cap (i.e. spiderweb graphs) or two caps.