Alexander numbering of knotted surface diagrams

Abstract

Abstract. A generic projection of a knotted oriented surface in 4-space divides 3-space into regions. The number of times (counted with sign) that a path from infinity to a given region intersects the projected surface is called the Alexander numbering of the region. The Alexander numbering is extended to branch and triple points of the projections. A formula that relates these indices is presented.

Twisted quandle homology theory and cocycle knot invariants

“…Proof By assumption there exists ξ ∈ C 1 TQ (X; N ) such that φ = δξ . By Equality (7), the map η ′ = sη − iξ gives rise to a desired quandle homomorphism.…”

“…Let α be an arc on the plane from a point in the region at infinity to a point R such that the interior of α misses all the crossing points of K and intersects transversely in finitely many points with the arcs of K . A classically known concept called Alexander numbering (see for example [11,7]) of R, denoted by L(R), is defined as the number, counted with signs, of the number of intersections between α and K . More specifically, when α is traced from the region at infinity to R, and intersect at p with K , if the normal to K at p is the same direction as α, then p contributes +1 to L(R).…”

Eigenvalue contour lines of Kac-Murdock-Szego matrices with a complex parameter

“…Evidently, our two rules are similar in nature to those used in computing winding numbers of closed, bounded, and oriented curves [8]. The latter rules, which appear in the literature in various contexts [10,11], are often referred to as Alexander numbering, the name originating from Alexander's 1928 paper [12]. More precisely, for the orientation provided in Fig.…”

The length of a 3–cocycle of the 5–dihedral quandle

“…See Figure 1. In particular, the X-set S D Z with the action s x D s C1 for any s 2 Z and x 2 X corresponds to an Alexander numbering [4], S D Z 2 with s x D s C 1 (mod 2) corresponds to a checkerboard coloring, and S D X corresponds to the original shadow coloring; see [3]. Let Col X ;S .D/ denote the set of .X; S/-colorings for D .…”