Abstract
The topological classification of nodal links and knot has enamored physicists and mathematicians alike, both for its mathematical elegance and implications on optical and transport phenomena. Central to this pursuit is the Seifert surface bounding the link/knot, which has for long remained a mathematical abstraction. Here we propose an experimentally realistic setup where Seifert surfaces emerge as boundary states of 4D topological systems constructed by stacking 3D nodal line systems along a 4th quasimomentum. We provide an explicit realization with 4D circuit lattices, which are freed from symmetry constraints and are readily tunable due to the dimension and distance agnostic nature of circuit connections. Importantly, their Seifert surfaces can be imaged in 3D via their pronounced impedance peaks, and are directly related to knot invariants like the Alexander polynomial and knot Signature. This work thus unleashes the great potential of Seifert surfaces as sophisticated yet accessible tools in exotic bandstructure studies.
Highlights
The topological classification of nodal links and knot has enamored physicists and mathematicians alike, both for its mathematical elegance and implications on optical and transport phenomena
We propose to realize 3D nodal loops (NLs) embedded in parent 4D nodal structures, such that Seifert surfaces naturally emerge as topologically robust zero-energy surfaces at their 3D boundaries
The combination of these topological invariants serves to precisely identify the nodal topology an approach to realize rather arbitrary linkages and knot topologies in momentum-space NLs defined in an effective 4D space, but still implementable via practical experimental settings in 3D
Summary
Under w^-direction OBCs, topological boundary states must appear due to the bulk-edge correspondence associated with a nontrivial Chern number, as shown in the Supplementary Note 1 and Fig. 1 Those boundary states at zero energy make up the Seifert surface cos kx þ cos ky< m À 2, kz 1⁄4 0 matching the identified NL (blue). The latter NL system has the curious property that each pair of loops is unlinked, even though the nodal structure has a nontrivial linkage characterized by the Milnor number[15]. While two different realizations of the same nodal knot/ link can have different reconstructed Seifert surfaces and different Seifert matrices/genera, their resultant nodal topological invariants (Alexander polynomial and signature) always have to agree
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