Knot polynomials of open and closed curves

Abstract

In this manuscript, we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves. We define the bracket polynomial of curves in 3-space and show that it has real coefficients and is a continuous function of the curve coordinates. This is used to define the Jones polynomial in a way that it is applicable to both open and closed curves in 3-space. For open curves, the Jones polynomial has real coefficients and it is a continuous function of the curve coordinates and as the endpoints of the curve tend to coincide, the Jones polynomial of the open curve tends to that of the resulting knot. For closed curves, it is a topological invariant, as the classical Jones polynomial. We show how these measures attain a simpler expression for polygonal curves and provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges.

Vassiliev measures of complexity of open and closed curves in 3-space

“…where K i is a knotoid that appears in a projection of l and p(K i ) is the geometric probability that the projection of l gives the knotoid K i and K(l) is the set of possible knotoids that can result as a projection of l. In [51], it was shown that p(K i ) is a continuous function of the curve coordinates. Thus, w k (l n ) is a continuous function of the coordinates of l n .…”

“…We can express p j 1 ,j 2 ,j 3 ,j 4 as the joint probability that e j 1 , e j 3 and e j 2 , e j 4 both cross. In [51], it was proved that the geometric probability that e j 1 , e j 3 cross, p j 1 ,j 3 , and the geometric probability that e j 2 , e j 4 cross, p j 2 ,j 4 , are continuous and are equal to the areas of the corresponding quadrangles on the sphere. Their intersection, p j 1 ,j 2 ,j 3 ,j 4 , is the area of the intersection of the two spherical quadrangles.…”

“…However, all these approaches focused on approximating the open curve in 3-space by a knot (a simple closed curve in 3-space) or by a knotoid (a two-dimensional diagram of an open-ended knot). In 2020 [51], the Jones polynomial of open curves in 3-space was introduced, and it was shown that it is a polynomial with real coefficients that are continuous functions of the curve coordinates. Therein, it was shown that the Jones polynomial of open curves in 3-space generalizes the conventional Jones polynomial.…”

“…Therein, it was shown that the Jones polynomial of open curves in 3-space generalizes the conventional Jones polynomial. In other words, the conventional Jones polynomial expression is a special case of the Jones polynomial introduced in [51].…”

“…The approach introduced in [51] provided a framework that we can use to study entanglement of both open and closed curves. This paper focuses on deriving Vassiliev invariant-type measures of entanglement of both open and closed curves.…”