Knot polynomials of open and closed curves Panagiotou, Eleni; Kauffman, Louis H.
Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences,
08/2020, Volume:
476, Issue:
2240
Journal Article
Peer reviewed
Open access
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.
In this paper, we derive formulae for the determinant of weaving knots W(3,n) and W(p,2). We calculate the dimension of the first homology group with coefficients in Z3 of the double cyclic cover of ...the 3-sphere S3 branched over W(3,n) and W(p,2) respectively. As a consequence, we obtain a lower bound of the unknotting number of W(3,n) for certain values of n.
Generalized Fishburn numbers and torus knots Bijaoui, Colin; Boden, Hans U.; Myers, Beckham ...
Journal of combinatorial theory. Series A,
February 2021, 2021-02-00, Volume:
178
Journal Article
Peer reviewed
Open access
Andrews and Sellers recently initiated the study of arithmetic properties of Fishburn numbers. In this paper we prove prime power congruences for generalized Fishburn numbers. These numbers are the ...coefficients in the 1−q expansion of the Kontsevich-Zagier series Ft(q) for the torus knots T(3,2t), t≥2. The proof uses a strong divisibility result of Ahlgren, Kim and Lovejoy and a new “strange identity” for Ft(q).
In this article, we explore a polynomial invariant for Legendrian knots which is a natural extension of Jones polynomial for (topological) knots. To this end, a new type of skein relation is ...introduced for the front projections of Legendrian knots. Further, we give a categorification of the polynomial invariant for Legendrian knots which is a natural extension of Khovanov homology. The Thurston-Bennequin invariant of Legendrian knot appears naturally in the construction of the homology as the grade shift.
The constructions of the polynomial invariant and its categorification are natural in the sense that if we treat Legendrian knots as only knots (that is, we forget the geometry on the knots), then we recover the Jones polynomial and Khovanov homology respectively. In the end, we discuss strengths and limitations of these invariants.
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