Legendrian Hopf Links

We completely classify Legendrian realizations of the Hopf link, up to coarse equivalence, in the 3-sphere with any contact structure.

On Cayley graphs of {\bb Z}^4

The generating sets of {\bb Z}^4 have been enumerated which consist of integral four-dimensional vectors with components −1, 0, 1 and allow Cayley graphs without edge intersections in a straight-edge embedding in a four-dimensional Euclidean space. Owing to computational restrictions the valency of enumerated graphs has been fixed to 10. Up to isomorphism 58 graphs have been found and characterized by coordination sequences, shortest cycles and automorphism groups. To compute automorphism groups, a novel strategy is introduced that is based on determining vertex stabilizers from the automorphism group of a sufficiently large finite ball cut out from an infinite graph. Six exceptional, rather `dense' graphs have been identified which are locally isomorphic to a five-dimensional cubic lattice within a ball of radius 10. They could be built by either interconnecting interpenetrated three- or four-dimensional cubic lattices and therefore necessarily contain Hopf links between quadrangular cycles. As a consequence, a local combinatorial isomorphism does not extend to a local isotopy.

Projective objects and the modified trace in factorisable finite tensor categories

For ${\mathcal{C}}$ a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show:(1)${\mathcal{C}}$ always contains a simple projective object;(2)if ${\mathcal{C}}$ is in addition ribbon, the internal characters of projective modules span a submodule for the projective $\text{SL}(2,\mathbb{Z})$-action;(3)the action of the Grothendieck ring of ${\mathcal{C}}$ on the span of internal characters of projective objects can be diagonalised;(4)the linearised Grothendieck ring of ${\mathcal{C}}$ is semisimple if and only if ${\mathcal{C}}$ is semisimple.Results (1)–(3) remain true in positive characteristic under an extra assumption. Result (1) implies that the tensor ideal of projective objects in ${\mathcal{C}}$ carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of $S$-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular $S$-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there.

A novel [4 + 3] interpenetrated net containing 7-fold interlocking pseudo-helical chains and exceptional catenane-like motifs

The first example of a [4 + 3] 7-fold interpenetrating network was prepared, showing 7-fold interlocking pseudo-helical chains and a unique catenane-like motif with Hopf links.

The Kauffman Polynomials of Generalized Hopf Links

Following the recent work by Chan [1] and Morton and Hadji [2] on the Homflypt polynomials of some generalized Hopf links, we investigate the polynomials of generalized Hopf links. By studying the Kauffman skein module of the solid torus S1 × D2, we establish a similar skein map on the Kauffman skein module of S1 × D2 which has distinct eigenvalues. Furthermore we are able to calculate the Kauffman polynomials of some specific generalized Hopf links.