Related papers: Jones rational coincidences
We prove that for knots, the evaluation of the Jones polynomial at the sixth root of unity, as well as the evaluation of the $Q$-polynomial at the reciprocal of the golden ratio, are uniquely determined by the oriented homeomorphism type of…
A sequence $f_n(q)$ is $q$-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in $q$ and $q^n$. Our main theorems state that $q$-holonomicity is preserved under twisting, i.e., replacing $q$ by $\omega q$…
It has been argued based on electric-magnetic duality and other ingredients that the Jones polynomial of a knot in three dimensions can be computed by counting the solutions of certain gauge theory equations in four dimensions. Here, we…
Many structures in science, engineering, and art can be viewed as curves in 3-space. The entanglement of these curves plays a crucial role in determining the functionality and physical properties of materials. Many concepts in knot theory…
The Jones polynomial, discovered in 1984, is an important knot invariant in topology. Among its many connections to various mathematical and physical areas, it is known (due to Witten) to be intimately connected to Topological Quantum Field…
Continued fractions whose elements are polynomial sequences have been carefully studied mostly in the cases where the degree of the numerator polynomial is less than or equal to two and the degree of the denominator polynomial is less than…
We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order…
We discuss two realizations of the colored Jones polynomials of a knot, one from an unnoticed work of the second author in 1994 on quantum R-matrices at roots of unity obtained from solutions of the pentagon identity, and another one from…
We exhibit an infinite family of knots with the property that the first coefficient of the n-colored Jones polynomial grows linearly with n. This shows that the concept of stability and tail seen in the colored Jones polynomials of…
Using the vertex model approach for braid representations, we compute polynomials for spin-1 placed on hyperbolic knots up to 15 crossings. These polynomials are referred to as 3-colored Jones polynomials or adjoint Jones polynomials.…
We describe a normal surface algorithm that decides whether a knot, with known degree of the colored Jones polynomial, satisfies the Strong Slope Conjecture. We also discuss possible simplifications of our algorithm and state related open…
We show that there are infinitely many pairs of alternating pretzel knots whose Jones polynomials are identical.
Knots, links and entangled filaments appear in many physical systems of interest in biology and engineering. Classifying knots and measuring entanglement is of interest both for advancing knot theory, as well as for analyzing large data…
We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of `knot invariants', among…
The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but…
This paper is an introduction to rational tangles, rational knots and links and their applications to DNA. The paper can be read as an introduction to our more technical papers on rational tangles (math.GT/0311499) and on rational knots…
We study near-alternating links whose diagrams satisfy conditions generalized from the notion of semi-adequate links. We extend many of the results known for adequate knots relating their colored Jones polynomials to the topology of…
We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl…
This paper gives new and elementary combinatorial topological proofs of the classification of unoriented and oriented rational knots and links. These proofs are based on the known classification of alternating knots through flyping, and the…
Turning the skein relation for HOMFLY into a Fibonacci recurrence, we prove that there are only three rational specializations of HOMFLY polynomial: Alexander-Conway, Jones, and a new one. Using the recurrence relation, we find general and…