Hyperbolic nodal band structures and knot invariants

We extend the list of known band structure topologies to include a large family of 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, taking the forms of Turk’s head knots, 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.

Cochran's βi-invariants via twisted Whitney towers

We show that Cochran's invariants [Formula: see text] of a [Formula: see text]-component link [Formula: see text] in the [Formula: see text]-sphere can be computed as intersection invariants of certain 2-complexes in the [Formula: see text]-ball with boundary [Formula: see text]. These 2-complexes are special types of twisted Whitney towers, which we call Cochran towers, and which exhibit a new phenomenon: A Cochran tower of order [Formula: see text] allows the computation of the [Formula: see text] invariants for all [Formula: see text], i.e. simultaneous extraction of invariants from a Whitney tower at multiple orders. This is in contrast with the order [Formula: see text] Milnor invariants (requiring order [Formula: see text] Whitney towers) and consistent with Cochran's result that the [Formula: see text] are integer lifts of certain Milnor invariants.

Milnor invariants of clover links

Levine introduced clover links to investigate the indeterminacy of Milnor invariants of links. He proved that for a clover link, Milnor numbers of length up to [Formula: see text] are well-defined if those of length [Formula: see text] vanish, and that Milnor numbers of length at least [Formula: see text] are not well-defined if those of length [Formula: see text] survive. For a clover link [Formula: see text] with vanishing Milnor numbers of length [Formula: see text], we show that the Milnor number [Formula: see text] for a sequence [Formula: see text] is well-defined by taking modulo the greatest common divisor of the [Formula: see text], where [Formula: see text] is any proper subsequence of [Formula: see text] obtained by removing at least [Formula: see text] indices. Moreover, if [Formula: see text] is a non-repeated sequence of length [Formula: see text], the possible range of [Formula: see text] is given explicitly. As an application, we give an edge-homotopy classification of [Formula: see text]-clover links.