SimCenter and Department of Mathematics Colloquium
Friday, April 16, 2021 at 2:00 PM
Contact Holley-Beeland@utc.edu for Zoom details.
Knot polynomials and Vassiliev measures of open and closed curves in 3-space
Eleni Panagiotou (Department of Mathematics – The University of Tennessee at Chattanooga)
Abstract:Knots and links appear in our everyday life, such as our shoelaces, or polymer melts, proteins and DNA. Measuring the complexity of such knots has been a challenge for many decades, due to the fact that they have open ends and thus do not satisfy the conditions for mathematical knots. Mathematical knots can be characterized using knot and link polynomials. In this talk we introduce a method to measure entanglement of curves in 3-space that extends the notion of knot and link polynomials to open curves in 3-space. We define the Jones polynomial of curves in 3-space and show that for open curves, it has real coefficients and it is a continuous function of the curve coordinates and for closed curves, it is a topological invariant, as the classical Jones polynomial. We show the Jones polynomial can attain a simpler expression for polygonal curves and we provide a finite form for their computation in the case of polygonal curves of 3 and 4 edges. As the complexity of the curves in 3-space increases, the computation of the Jones polynomial increases as well. We introduce the Vassiliev measures of entanglement of open curves. We will show that these capture some of the information of the Jones polynomial with much less computational cost.