Classification of prime links in the thickened torus having crossing number 5

The goal of this paper is to tabulate all prime links in the thickened torus [Formula: see text] having diagrams having crossing number 5. First, we construct a table of prime projections of links on the torus [Formula: see text] having exactly 5 crossings. To this end, we enumerate abstract quadrivalent graphs of special type and consider all possible embeddings of the graphs into the torus [Formula: see text] in order to construct prime projections. Then, we prove that all obtained projections are inequivalent. Second, we use the list of prime projections to construct a table of diagrams of prime links in the torus [Formula: see text]. In order to prove that all those links are inequivalent, we use two modifications of the Kauffman bracket. Several known and new tricks allow us to keep the process within reasonable limits and rigorously theoretically prove the completeness of the constructed tables.

Gluing formulas for the L2-Alexander torsions

We prove a Torres-like formula for the [Formula: see text]-Alexander torsions of links, as well as formulas for connected sums and cablings of links. Along the way we compute explicitly the [Formula: see text]-Alexander torsions of torus links inside the three-sphere, the solid torus and the thickened torus.

Elliptic associators and the LMO functor

The elliptic associator of Enriquez can be used to define an invariant of tangles embedded in the thickened torus, which extends the Kontsevich integral. This construction by Humbert uses the formulation of categories with elliptic structures. In this work we show that an extension of the LMO functor also leads to an elliptic structure on the category of Jacobi diagrams which is used by the Kontsevich integral, and find the relation between the two structures. We use this relation to give an alternative proof for the properties of the elliptic associator of Enriquez. Those results can lead the way to finding associators for higher genera.

Classification of genus 1 virtual knots having at most five classical crossings

The goal of this paper is to tabulate all genus one prime virtual knots having diagrams with ≤ 5 classical crossings. First, we construct all nonlocal prime knots in the thickened torus T × I which have diagrams with ≤ 5 crossings and admit no destabilizations. Then we use a generalized version of the Kauffman polynomial to prove that all those knots are different. Finally, we convert the knot diagrams in T thus obtained into virtual knot diagrams in the plane.

DOUBLY PERIODIC TEXTILE STRUCTURES

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.