Speaker: Kasturi Barkataki

Location and Time: AC1 115, 2:10–2:40pm

Title: The Jones polynomial of collections of open curves in 3-space

Abstract: Measuring the entanglement complexity of collections of open curves in 3-space is a long standing open mathematical problem with many applications, especially in polymer and biopolymer systems. In this paper, we give a novel definition of the Jones polynomial that generalizes the classic Jones polynomial to collections of open curves in 3-space. More precisely, first we create the Jones polynomial of linkoids (open link diagrams) and show that this is a well-defined topological invariant. We show that for link-type linkoids, the Jones polynomial coincides with that of the corresponding link. The new definition of the Jones polynomial of linkoids enables us to define the Jones polynomial of collections of open and closed curves in 3-space by averaging over $S^2$ . For collections of open curves, the Jones polynomial has real coefficients and it is a continuous function of the curves’ coordinates. As the endpoints of each component of the system tend to coincide to form a link, the Jones polynomial tends to that of the resultant link.