An algorithm for computing the Seifert matrix of a link from a braid representation

Boundary states of 4D topological matter: Emergence and full 3D-imaging of nodal Seifert surfaces

“…Essentially, drumhead states are boundary projections of a bulk surface stretched across the NLs i.e. a taut Seifert surface [14,15] of the NLs, with degeneracy corresponding to the multiplicity of the projection. But it has to be emphasized that this Seifert surface of a 3D nodal system is an entirely abstract construction [47] not verifiable from 2D boundary states, insofar as 3D geometric information, particularly of the knot over/under-crossings, is already lost in the surface projection.…”

“…A quintessential example is given by nodal knots existing in momentum space, where the knotted structure leads to new phases of matter protected by topological knot invariants. Unlike knotted molecules or optical vortices in real space [1][2][3], nodal knots consist of valence and conduction bands intersecting along one-dimensional (1D) lines in momentum space, which intertwine to form knotted nodal loops (NLs) [4][5][6][7][8][9][10][11][12][13] so multifarious that topological invariants take the form of polynomials rather than the Z 2 or Z integers [11][12][13][14][15] of ordinary topological insulators. Fundamental in constructing such invariants are the Seifert surfaces bounded by the nodal structure [14,15], which assume interesting, bubble-like shapes demarcating "drumhead" topological regions in the projected 2D surface Brillouin zone (BZ) [16][17][18][19].…”

“…As compact and orientable surfaces bounded by nodal knots or links, Seifert surfaces not only provide convenient visualization, but are also of core importance to topological classification. The linking properties of their homology generators can be used to compute [11,15] the Alexander polynomial -a classical knot invariant -of the NL or knot, hence distinguishing it from other nodal configurations. Indeed, Seifert surfaces are central to knot theory and low-dimensional topology [20], provoking many fascinating mathematical and computational problems, such as the uniqueness of a minimal genus Seifert surface [21], and their construction and visualization [22,23].…”

Realistic Floquet Semimetal with Exotic Topological Linkages between Arbitrarily Many Nodal Loops

“…In this standard form, it is easy to define a homology basis (green) consisting of the independent closed loops traversing the twist operators. From this wedding cake, it is also not difficult to deduce the a representative braid for the knot/link by identifying the twist operators as braid operations, and the layers of the cake as the strands [62]. 2a, together with its lifted l # .…”

Signatures in algebra, topology and dynamics

“…We refer to Stallings [152], Gambaudo and Ghys [57], Cohen and van Wijk [41], Collins [42], Bourrigan [31] and Palmer [125] for more detailed accounts of the Seifert surfaces and Seifert matrices of braids.…”