Fig. 1: Nodal knots from braids. | Nature Communications

Fig. 1: Nodal knots from braids.

From: Imaging nodal knots in momentum space through topolectrical circuits

Fig. 1

a Braid operations σi and \({\sigma }_{i}^{-1}\) represent the over/under-crossing of strand i with strand i + 1 as we travel upwards. A braid consists of a series of braid operations, and can be closed to form a knot or link (in this case it is a link between three loops). b A braid closure can be embedded onto the 3D BZ torus in different ways through different choices of F(k). Depending on its topological charge density distribution of Eq. (4), it can produce different numbers of copies of the knots in the BZ, i.e. one a single copy (F1) or two mirror imaged copies (F2). cf Various examples of simple Nodal knots/links defined by Eq. (3), some of which we shall explicitly construct in circuits band structures later. c Hopf-link with \(\sigma ={\sigma }_{1}^{2}\) and f(zw) = (z − w)(z + w). d) Trefoil knot with \(\sigma ={\sigma }_{1}^{3}\) and f(zw) = (z − w3/2)(z + w3/2). e) 3-link with \(\sigma ={({\sigma }_{1}{\sigma }_{2}{\sigma }_{1})}^{2}\) and f(zw) = z(z2 − w2). f Figure-8 knot with \(\sigma ={({\sigma }_{2}^{-1}{\sigma }_{1})}^{2}\) and \(f(z,w)=64{z}^{3}-12z(3+2({w}^{2}-{\bar{w}}^{2}))-14({w}^{2}+{\bar{w}}^{2})-({w}^{4}-{\bar{w}}^{4})\)35.

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