Hyperbolic Nodal Band Structures and Knot Invariants. (arXiv:1905.05858v1 [cond-mat.mes-hall])

We extend the list of known band structure topologies to include hyperbolic
nodal links and knots, occurring both in conventional Hermitian systems where
their stability relies on discrete symmetries, and in the dissipative
non-Hermitian realm where the knotted nodal lines are generic and thus stable
towards any small perturbation. We show that these nodal structures, including
the figure-eight knot and the Borromean rings, appear in both continuum- and
lattice models with relatively short-ranged hopping that is within experimental
reach. To determine the topology of the nodal structures, we devise an
efficient algorithm for computing the Alexander polynomial, linking numbers and
higher order Milnor invariants based on an approximate and well controlled
parameterisation of the knot.

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