Fig. 2: Seifert surfaces from topological surface states. | Nature Communications

Fig. 2: Seifert surfaces from topological surface states.

From: Imaging nodal knots in momentum space through topolectrical circuits

Fig. 2

Projected surface states on the (001) surface of the a Hopf-link with \(\sigma ={\sigma }_{1}^{2}\), b Borromean rings with \(\sigma ={({\sigma }_{2}^{-1}{\sigma }_{1})}^{3}\) and c) 3-link with \(\sigma ={({\sigma }_{1}{\sigma }_{2}{\sigma }_{1})}^{2}\). We can observe multiple folded layers of the surface on top of another. Note that a different parametrization was used to plot these surfaces, as compared to Fig. 1. Interestingly, b, c, both contain three loops, but b is totally unlinked upon removal of any single loop, while, c still reduces to a Hopf-link upon removal of any loop. d How a Seifert surface can be obtained from the Drumhead states. By comparing the same nodal crossings across Drumhead states from different surfaces (Left), one can deduce the over/under-crossings in a knot diagram. The interior of this knot can then be systematically promoted into “surface layers” bounded by appropriately defined crossings (Center), which can further be arranged into a layer arrangement where its homology loops (i.e., α1) are evident.

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