More Like this

1. Knot polynomials of open and closed curves

   https://doi.org/10.1098/rspa.2020.0124  

   Panagiotou, Eleni; Kauffman, Louis H. (August 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   null (Ed.)

   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. 

   more » « less
2. Vassiliev measures of complexity of open and closed curves in 3-space

   https://doi.org/10.1098/rspa.2021.0440  

   Panagiotou, Eleni; Kauffman, Louis H. (October 2021, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   In this article, we define Vassiliev measures of complexity for open curves in 3-space. These are related to the coefficients of the enhanced Jones polynomial of open curves in 3-space. These Vassiliev measures are continuous functions of the curve coordinates; as the ends of the curve tend to coincide, they converge to the corresponding Vassiliev invariants of the resulting knot. We focus on the second Vassiliev measure from the enhanced Jones polynomial for closed and open curves in 3-space. For closed curves, this second Vassiliev measure can be computed by a Gauss code diagram and it has an integral formulation, the double alternating self-linking integral. The double alternating self-linking integral is a topological invariant of closed curves and a continuous function of the curve coordinates for open curves in 3-space. For polygonal curves, the double alternating self-linking integral obtains a simpler expression in terms of geometric probabilities. 

   more » « less
3. The Jones polynomial in systems with periodic boundary conditions

   https://doi.org/10.1088/1751-8121/ad36fe  

   Barkataki, Kasturi; Panagiotou, Eleni (April 2024, Journal of Physics A: Mathematical and Theoretical)

   Abstract Entanglement of collections of filaments arises in many contexts, such as in polymer melts, textiles and crystals. Such systems are modeled using periodic boundary conditions (PBCs), which create an infinite periodic system whose global entanglement may be impossible to capture and is repetitive. We introduce two new methods to assess topological entanglement in PBC: the Periodic Jones polynomial and the Cell Jones polynomial. These tools capture the grain of geometric/topological entanglement in a periodic system of open or closed chains, by using a finite link as a representative of the global system. These polynomials are topological invariants in some cases, but in general are sensitive to both the topology and the geometry of physical systems. For a general system of 1 closed chain in 1 PBC, we prove that the Periodic Jones polynomial is a recurring factor, up to a remainder, of the Jones polynomial of a conveniently chosen finite cutoff of arbitrary size of the infinite periodic system. We apply the Cell Jones polynomial and the Periodic Jones polynomial to physical PBC systems such as 3D realizations of textile motifs and polymer melts of linear chains obtained from molecular dynamics simulations. Our results demonstrate that the Cell Jones polynomial and the Periodic Jones polynomial can measure collective geometric/topological entanglement complexityin such systems of physical relevance. 

   more » « less
4. The second Vassiliev measure of uniform random walks and polygons in confined space

   https://doi.org/10.1088/1751-8121/ac4abf  

   Smith, Philip; Panagiotou, Eleni (February 2022, Journal of Physics A: Mathematical and Theoretical)

   Abstract Biopolymers, like chromatin, are often confined in small volumes. Confinement has a great effect on polymer conformations, including polymer entanglement. Polymer chains and other filamentous structures can be represented by polygonal curves in three-space. In this manuscript, we examine the topological complexity of polygonal chains in three-space and in confinement as a function of their length. We model polygonal chains by equilateral random walks in three-space and by uniform random walks (URWs) in confinement. For the topological characterization, we use the second Vassiliev measure. This is an integer topological invariant for polygons and a continuous functions over the real numbers, as a function of the chain coordinates for open polygonal chains. For URWs in confined space, we prove that the average value of the Vassiliev measure in the space of configurations increases as O ( n 2 ) with the length of the walks or polygons. We verify this result numerically and our numerical results also show that the mean value of the second Vassiliev measure of equilateral random walks in three-space increases as O ( n ). These results reveal the rate at which knotting of open curves and not simply entanglement are affected by confinement. 

   more » « less
5. Quasipositive links and Stein surfaces

   https://doi.org/10.2140/gt.2021.25.1441  

   Hayden, Kyle (January 2021, Geometry & Topology)

   We study the generalization of quasipositive links from the 3-sphere to arbitrary closed, orientable 3-manifolds. Our main result shows that the boundary of any smooth, properly embedded complex curve in a Stein domain is a quasipositive link. This generalizes a result due to Boileau and Orevkov, and it provides the first half of a topological characterization of links in 3-manifolds which bound complex curves in a Stein filling. Our arguments replace pseudoholomorphic curve techniques with a study of characteristic and open book foliations on surfaces in 3- and 4-manifolds. 

   more » « less