**SimCenter and Department of Mathematics Colloquium**

**Friday, April 23, 2021 at 2:00 PM**

Contact Holley-Beeland@utc.edu for Zoom details.

### The Second Vassiliev measure of polygonal curves in 3-space

**Philip Smith (***Student, Department of Mathematics – the University of Tennessee at Chattanooga*)

*Student, Department of Mathematics – the University of Tennessee at Chattanooga*)

**Abstract: **Open knots and links are present in many physical systems, such as polymers and biopolymers. These physical filaments can be represented by polygonal chains in 3-space, whose complexity can be studied by using tools from Knot Theory. A second Vassiliev invariant, the Casson invariant, is a measure of topological complexity which can distinguish knots and links, as knot and link polynomials do, but it is easier to calculate. Recently, the second Vassiliev measure was defined for open curves in 3-space. For open curves, this is a measure of higher order complexity that is a continuous function of the chain coordinates. We apply the Casson measure to polygonal curves of varying length to examine how their complexity varies as a function of length. Our results will be used to quantify entanglement complexity in proteins and other polymers.

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