Narrow Results

We introduce a map, which we call the bind map, from the set of all classical braids to that of all trivalent braids. Then we define a notation for handlebody-links with a pair of a bind map and a classical braid. We show that isotopies and braid relations are equivalent for trivial handlebody-braids obtained from classical 3-braids with bind maps. We introduce two types of graphs for a classical braid, which indicate how the binding forms of the trivalent braids are transformed each other. We determine all patterns of the graphs for classical 3-braids.

Spin interaction Hamiltonians are obtained from the unitary Yang–Baxter Ř-matrix. Based on which, we study Berry phase and quantum criticality in the Yang–Baxter systems. We also indicate that Bogoliubov Hamiltonian is a derivative of Dirac Hamiltonian via braid relation.