Differential Geometry and Topology of Curves 2024, 43: Alexander's Polynomial

The knot group is an invariant, though two arbitrary isomorphic groups are difficult to determine. A knot can be considered a simple, closed, directed polygon in 3D Euclidean space. On a knot, one may generally subdivide an edge into subedges by creating a new vertex at a point on it, do the same in reverse, and change the shape of the knot by continuously displacing a vertex so that the knot never acquires a singularity. Two knots are of the same type, iff one of them is transformable into the other by a finite succession of these elementary operations. Oriented knots can be viewed as diagrams where all self-intersection points show up as vertices, which divide the diagram into its regions. At a self-intersection point, two of the four corners are dotted along the lower branch. Each region has an index, the first of which is chosen arbitrarily, and the other ones being chosen off this first one. One crosses the curves from right to left, passing from a region of index p to a region of next higher index p + 1. At all self-intersections there are two opposite corners of the same index and two opposite corners of index p - 1, and p + 1. p is the index of the self-intersection point. The diagram in this way spans up a system of equations for the diagram. They have the form of Alexander's Polynomial.

When the resulting matrix can be reduced to a square matrix by striking out two columns corresponding to regions with consecutive indices, the determinant of the residual matrix will be independent of the two struck out columns to a factor of the form ±xⁿ.

If two diagrams represent knots of the same type, the corresponding matrices are ε-equivalent. An elementary curve in n-dimensional Euclidean space is the image of the segment [a, b] under a homeomorphism, so each elementary curve has a parametric representation.

with vector-functions f(t). Cᵏ-regular curves have parametric representations with non-zero coordinate functions of class Cᵏ. In the case that f is indexed k, then the curve is smooth. The length s of a curve is a supremum of lengths of polygonal lines. If it's well-defined, the curve is "rectifiable". The curvatures are derived from the unit tangent vector.

If the curvature functions are continuous, there is a C²-regular curve (unique up to a rigid motion), in n-dimensional Euclidean space, having the curvature functions as its own curvature, and s as its arclength.