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1. Twist number and the alternating volume of knots

   https://doi.org/10.1142/S0218216519500160  

   Blair, Ryan; Allen, Heidi; Rodriguez, Leslie (January 2019, Journal of Knot Theory and Its Ramifications)

   It was previously shown by the first author that every knot in [Formula: see text] is ambient isotopic to one component of a two-component, alternating, hyperbolic link. In this paper, we define the alternating volume of a knot [Formula: see text] to be the minimum volume of any link [Formula: see text] in a natural class of alternating, hyperbolic links such that [Formula: see text] is ambient isotopic to a component of [Formula: see text]. Our main result shows that the alternating volume of a knot is coarsely equivalent to the twist number of a knot. 

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2. The Jones polynomial of collections of open curves in 3-space

   https://doi.org/10.1098/rspa.2022.0302  

   Barkataki, Kasturi; Panagiotou, Eleni (November 2022, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Measuring the entanglement complexity of collections of open curves in 3-space has been an intractable, yet pressing mathematical problem, relevant to a plethora of physical systems, such as in polymers and biopolymers. In this manuscript, 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 provide a novel definition of the Jones polynomial of linkoids (open link diagrams) and show that this is a well-defined single variable polynomial that is a topological invariant, which, for link-type linkoids, coincides with that of the corresponding link. Using the framework introduced in (Panagiotou E, Kauffman L. 2020 Proc. R. Soc. A 476 , 20200124. (( doi:10.1098/rspa.2020.0124 )), this enables us to define the Jones polynomial of collections of open and closed curves in 3-space. For collections of open curves in 3-space, the Jones polynomial has real coefficients and it is a continuous function of the curves’ coordinates. As the endpoints of the curves tend to coincide, the Jones polynomial tends to that of the resultant link. We demonstrate with numerical examples that the novel Jones polynomial enables us to characterize the topological/geometrical complexity of collections of open curves in 3-space for the first time. 

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3. Log-Concavity of the Alexander Polynomial

   https://doi.org/10.1093/imrn/rnae058  

   Hafner, Elena S; Mészáros, Karola; Vidinas, Alexander (April 2024, International Mathematics Research Notices)

   Abstract The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox’s conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial $$\Delta _{L}(t)$$ of an alternating link $$L$$ are unimodal. Fox’s conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsváth and Szabó (2003) for the case of genus $$2$$ alternating knots, among others. We settle Fox’s conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of $$\Delta _{L}(t)$$, where $$L$$ is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal. 

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4. A new Gram determinant from the Möbius band

   https://doi.org/10.1142/S0218216524500378  

   Ibarra, Dionne; Montoya-Vega, Gabriel (October 2024, Journal of Knot Theory and Its Ramifications)

   Gram determinants earned traction among knot theorists after Witten’s presumption about the existence of a 3-manifold invariant connected to the Jones polynomial. Triggered by the creation of such an invariant by Reshetikhin and Turaev, several mathematicians have explored this line of research ever since. Gram determinants came into play by Lickorish’s skein theoretic approach to the invariant. The construction of different bilinear forms is possible through changes in the ambient surface of the Kauffman bracket skein module. Hence, different types of Gram determinants have arisen in knot theory throughout the years; some of these determinants are discussed here. In this paper, we introduce a new version of such a determinant from the Möbius band and prove some important results about its structure. In particular, we explore its connection to the annulus case and factors of its closed formula. 

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5. 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. 

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