knot polynomial
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2020 ◽  
Vol 3 (1) ◽  
Author(s):  
Xin Liu ◽  
Renzo L. Ricca ◽  
Xin-Fei Li

2018 ◽  
Vol 147 (1) ◽  
pp. 377-397
Author(s):  
Dror Bar-Natan ◽  
Roland van der Veen

2018 ◽  
Vol 33 (27) ◽  
pp. 1850156 ◽  
Author(s):  
Xinfei Li ◽  
Xin Liu

Superconducting cosmic strings (SCSs) have received revived interests recently. In this paper we treat closed SCSs as oriented knotted line defects, and concentrate on their topology by studying the Hopf topological invariant. This invariant is an Abelian Chern–Simons action, from which the HOMFLYPT knot polynomial can be constructed. It is shown that the two independent parameters of the polynomial correspond to the writhe and twist contributions, separately. This new method is topologically stronger than the traditional (self-)linking number method, which fails to detect essential topology of knots sometimes, shedding new light upon the study of physical intercommunications of superconducting cosmic strings as a complex system.


2017 ◽  
Vol 32 (27) ◽  
pp. 1750164 ◽  
Author(s):  
Xinfei Li ◽  
Xin Liu ◽  
Yong-Chang Huang

Cosmic strings in the early universe have received revived interest in recent years. In this paper, we derive these structures as topological defects from singular distributions of the quintessence field of dark energy. Our emphasis is placed on the topological charge of tangled cosmic strings, which originates from the Hopf mapping and is a Chern–Simons action possessing strong inherent tie to knot topology. It is shown that the Kauffman bracket knot polynomial can be constructed in terms of this charge for unoriented knotted strings, serving as a topological invariant much stronger than the traditional Gauss linking numbers in characterizing string topology. Especially, we introduce a mathematical approach of breaking-reconnection which provides a promising candidate for studying physical reconnection processes within the complexity-reducing cascades of tangled cosmic strings.


2017 ◽  
Vol 5 (1) ◽  
pp. 21-30 ◽  
Author(s):  
Wei Tian ◽  
Xue Lei ◽  
Louis H. Kauffman ◽  
Jie Liang

Abstract Knot polynomials have been used to detect and classify knots in biomolecules. Computation of knot polynomials in DNA and protein molecules have revealed the existence of knotted structures, and provided important insight into their topological structures. However, conventional knot polynomials are not well suited to study RNA molecules, as RNA structures are determined by stem regions which are not taken into account in conventional knot polynomials. In this study, we develop a new class of knot polynomials specifically designed to study RNA molecules, which considers stem regions. We demonstrate that our knot polynomials have direct structural relation with RNA molecules, and can be used to classify the topology of RNA secondary structures. Furthermore, we point out that these knot polynomials can be used to model the topological effects of disulfide bonds in protein molecules.


2015 ◽  
Vol 24 (08) ◽  
pp. 1550047 ◽  
Author(s):  
Tomoko Yanagimoto

The zeroth coefficient polynomial of the skein (HOMFLYPT) knot polynomial called the Γ-polynomial is studied from a viewpoint of regular homotopy of knot diagrams. In particular, an elementary existence proof of the knot invariance of the Γ-polynomial is given. After observing that there are three types for 2-string tangle diagrams, the Γ-polynomial is generalized to a polynomial invariant of a 2-string tangle. As an application, we have a new proof of the assertion that Kinoshita's θ-curve is not equivalent to the trivial θ-curve.


2013 ◽  
Vol 22 (10) ◽  
pp. 1350060 ◽  
Author(s):  
JONATHAN GRANT

Motivated by a possible connection between the SU (N) instanton knot Floer homology of Kronheimer and Mrowka and 𝔰𝔩(N) Khovanov–Rozansky homology, Lobb and Zentner recently introduced a moduli problem associated to colorings of trivalent graphs of the kind considered by Murakami, Ohtsuki and Yamada in their state-sum interpretation of the quantum 𝔰𝔩(N) knot polynomial. For graphs with two colors, they showed this moduli space can be thought of as a representation variety, and that its Euler characteristic is equal to the 𝔰𝔩(N) polynomial of the graph evaluated at 1. We extend their results to graphs with arbitrary colorings by irreducible anti-symmetric representations of 𝔰𝔩(N).


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