scholarly journals On Some Moves on Links and the Hopf Crossing Number

2020 ◽  
Vol 18 (1) ◽  
Author(s):  
Maciej Mroczkowski
Keyword(s):  
Primitive Root ◽  
Jones Polynomial ◽  
Crossing Number ◽  
Equivalence Classes ◽  
Lens Spaces ◽  
Root Of Unity ◽  
Arrow Diagrams

AbstractWe consider arrow diagrams of links in $$S^3$$ S 3 and define k-moves on such diagrams, for any $$k\in \mathbb {N}$$ k ∈ N . We study the equivalence classes of links in $$S^3$$ S 3 up to k-moves. For $$k=2$$ k = 2 , we show that any two knots are equivalent, whereas it is not true for links. We show that the Jones polynomial at a k-th primitive root of unity is unchanged by a k-move, when k is odd. It is multiplied by $$-1$$ - 1 , when k is even. It follows that, for any $$k\ge 5$$ k ≥ 5 , there are infinitely many classes of knots modulo k-moves. We use these results to study the Hopf crossing number. In particular, we show that it is unbounded for some families of knots. We also interpret k-moves as some identifications between links in different lens spaces $$L_{p,1}$$ L p , 1 .

Entanglement complexity of graphs in Z3

1992 ◽  
Vol 111 (1) ◽  
pp. 75-91 ◽  
Author(s):  
C. E. Soteros ◽  
D. W. Sumners ◽  
S. G. Whittington
Keyword(s):  
Crossing Number ◽  
Equivalence Classes ◽  
Closed Curve ◽  
Good Measure ◽  
Average Value ◽  
Cut Edge ◽  
Expected Complexity ◽  
Almost All

AbstractIn this paper we are concerned with questions about the knottedness of a closed curve of given length embedded in Z3. What is the probability that such a randomly chosen embedding is knotted? What is the probability that the embedding contains a particular knot? What is the expected complexity of the knot? To what extent can these questions also be answered for a graph of a given homeomorphism type?We use a pattern theorem due to Kesten 12 to prove that almost all embeddings in Z3 of a sufficiently long closed curve contain any given knot. We introduce the idea of a good measure of knot complexity. This is a function F which maps the set of equivalence classes of embeddings into 0, ). The F measure of the unknot is zero, and, generally speaking, the more complex the prime knot decomposition of a given knot type, the greater its F measure. We prove that the average value of F diverges to infinity as the length (n) of the embedding goes to infinity, at least linearly in n. One example of a good measure of knot complexity is crossing number.Finally we consider similar questions for embeddings of graphs. We show that for a fixed homeomorphism type, as the number of edges n goes to infinity, almost all embeddings are knotted if the homeomorphism type does not contain a cut edge. We prove a weaker result in the case that the homeomorphism type contains at least one cut edge and at least one cycle.

scholarly journals The Jones polynomial: quantum algorithms and applications in quantum complexity theory

2008 ◽  
Vol 8 (1&2) ◽  
pp. 147-180
Author(s):  
P. Wocjan ◽  
J. Yard
Keyword(s):  
Quantum Computation ◽  
Braid Group ◽  
Quantum Algorithms ◽  
Primitive Root ◽  
Jones Polynomial ◽  
Tutte Polynomial ◽  
Roots Of Unity ◽  
Complete Problems ◽  
Primitive Roots ◽  
Quantum Complexity

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 representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al.\ that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a \#P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

THE (2, ∞)-SKEIN MODULE OF LENS SPACES: A GENERALIZATION OF THE JONES POLYNOMIAL

1993 ◽  
Vol 02 (03) ◽  
pp. 321-333 ◽  
Author(s):  
JIM HOSTE ◽  
JÓZEF H. PRZYTYCKI
Keyword(s):  
Jones Polynomial ◽  
Lens Spaces ◽  
Skein Module

We extend the Jones polynomial for links in S3 to links in L(p, q), p>0. Specifically, we show that the (2, ∞)-skein module of L(p, q) is free with [p/2]+1 generators. In the case of S1×S2 the skein module is infinitely generated.

scholarly journals Estimating Jones and HOMFLY polynomials with one clean qubit

2009 ◽  
Vol 9 (3&4) ◽  
pp. 264-289
Author(s):  
S.P. Jordan ◽  
P. Wocjan
Keyword(s):  
Quantum Computing ◽  
Quantum Computer ◽  
Jones Polynomial ◽  
Single Variable ◽  
Initial State ◽  
Link Invariants ◽  
Root Of Unity ◽  
Complete Problem ◽  
The One ◽  
Homfly Polynomials

The Jones and HOMFLY polynomials are link invariants with close connections to quantum computing. It was recently shown that finding a certain approximation to the Jones polynomial of the trace closure of a braid at the fifth root of unity is a complete problem for the one clean qubit complexity class\cite{Shor_Jordan}. This is the class of problems solvable in polynomial time on a quantum computer acting on an initial state in which one qubit is pure and the rest are maximally mixed. Here we generalize this result by showing that one clean qubit computers can efficiently approximate the Jones and single-variable HOMFLY polynomials of the trace closure of a braid at \emph{any} root of unity.

A New Class of Representations of EALA Coordinated by Quantum Tori in Two Variables

2002 ◽  
Vol 45 (4) ◽  
pp. 672-685 ◽  
Author(s):  
S. Eswara Rao ◽  
Punita Batra
Keyword(s):  
Lie Algebras ◽  
Primitive Root ◽  
Irreducible Module ◽  
Affine Lie Algebras ◽  
Quantum Torus ◽  
New Class ◽  
Root Of Unity ◽  
Quantum Tori ◽  
Finite Dimensional

AbstractWe study the representations of extended affine Lie algebras where q is N-th primitive root of unity (ℂq is the quantum torus in two variables). We first prove that ⊕ for a suitable number of copies is a quotient of . Thus any finite dimensional irreducible module for ⊕ lifts to a representation of . Conversely, we prove that any finite dimensional irreducible module for comes from above. We then construct modules for the extended affine Lie algebras which is integrable and has finite dimensional weight spaces.

DEGREES OF THE JONES POLYNOMIALS OF CERTAIN PRETZEL LINKS

2000 ◽  
Vol 09 (07) ◽  
pp. 907-916 ◽  
Author(s):  
MASAO HARA ◽  
SEI'ICHI TANI ◽  
MAKOTO YAMAMOTO
Keyword(s):  
Nonnegative Integer ◽  
Jones Polynomial ◽  
Crossing Number ◽  
Kauffman Bracket ◽  
Jones Polynomials ◽  
Bracket Polynomials

We calculate the highest and the lowest degrees of the Kauffman bracket polynomials of certain inadequate pretzel links and show that there is a knot K such that c(K) – r _ deg VK=k for any nonnegative integer k, where c(K) is the crossing number of K and r _ deg VK is the reduced degree of the Jones polynomial VK of K.

scholarly journals Cyclotomic analogues of finite multiple zeta values

2018 ◽  
Vol 154 (12) ◽  
pp. 2701-2721 ◽  
Author(s):  
Henrik Bachmann ◽  
Yoshihiro Takeyama ◽  
Koji Tasaka
Keyword(s):  
Vector Space ◽  
Primitive Root ◽  
Large Degree ◽  
Multiple Zeta Values ◽  
Linear Relations ◽  
Multiple Zeta Value ◽  
Root Of Unity ◽  
Multiple Zeta ◽  
Zeta Values

We study the values of finite multiple harmonic $q$-series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetric multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we prove the duality formula for these values, as an example of linear relations, which induce those among FMZVs and SMZVs simultaneously. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic $q$-series at a primitive root of unity of sufficiently large degree.

scholarly journals Crosscap number of knots and volume bounds

2020 ◽  
Vol 31 (13) ◽  
pp. 2050111
Author(s):  
Noboru Ito ◽  
Yusuke Takimura
Keyword(s):  
Jones Polynomial ◽  
Upper Bounds ◽  
Crossing Number ◽  
Proved Theorem ◽  
Knot Invariants ◽  
Chord Diagrams ◽  
Hyperbolic Volume ◽  
Knot Invariant ◽  
Twist Number

In this paper, we obtain the crosscap number of any alternating knots by using our recently-introduced diagrammatic knot invariant (Theorem 1). The proof is given by properties of chord diagrams (Kindred proved Theorem 1 independently via other techniques). For non-alternating knots, we give Theorem 2 that generalizes Theorem 1. We also improve known formulas to obtain upper bounds of the crosscap number of knots (alternating or non-alternating) (Theorem 3). As a corollary, this paper connects crosscap numbers and our invariant with other knot invariants such as the Jones polynomial, twist number, crossing number, and hyperbolic volume (Corollaries 1–7). In Appendix A, using Theorem 1, we complete giving the crosscap numbers of the alternating knots with up to 11 crossings including those of the previously unknown values for [Formula: see text] knots (Tables A.1).

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