scholarly journals ASYMPTOTICS OF THE QUANTUM INVARIANTS FOR SURGERIES ON THE FIGURE 8 KNOT

2006 ◽  
Vol 15 (04) ◽  
pp. 479-548 ◽  
Author(s):  
JØRGEN ELLEGAARD ANDERSEN ◽  
SØREN KOLD HANSEN
Keyword(s):  
Jones Polynomial ◽  
Saddle Point Method ◽  
Stationary Points ◽  
Quantum Invariants ◽  
Contour Integrals ◽  
Volume Conjecture ◽  
Phase Functions ◽  
Chern Simons ◽  
Large Quantum ◽  
Double Contour

We investigate the Reshetikhin–Turaev invariants associated to SU(2) for the 3-manifolds M obtained by doing any rational surgery along the figure 8 knot. In particular, we express these invariants in terms of certain complex double contour integrals. These integral formulae allow us to propose a formula for the leading asymptotics of the invariants in the limit of large quantum level. We analyze this expression using the saddle point method. We construct a certain surjection from the set of stationary points for the relevant phase functions onto the space of conjugacy classes of nonabelian SL(2, ℂ)-representations of the fundamental group of M and prove that the values of these phase functions at the relevant stationary points equals the classical Chern–Simons invariants of the corresponding flat SU(2)-connections. Our findings are in agreement with the asymptotic expansion conjecture. Moreover, we calculate the leading asymptotics of the colored Jones polynomial of the figure 8 knot following Kashaev [14]. This leads to a slightly finer asymptotic description of the invariant than predicted by the volume conjecture [24].

On the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings

2017 ◽  
Vol 165 (2) ◽  
pp. 287-339 ◽  
Author(s):  
TOMOTADA OHTSUKI ◽  
YOSHIYUKI YOKOTA
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Point ◽  
Asymptotic Expansions ◽  
Saddle Point Method ◽  
Hyperbolic Structure ◽  
3 Dimensional ◽  
Volume Conjecture ◽  
Hyperbolic Volume ◽  
Chern Simons ◽  
Dimensional Domain

AbstractWe give presentations of the asymptotic expansions of the Kashaev invariant of the knots with 6 crossings. In particular, we show the volume conjecture for these knots, which states that the leading terms of the expansions present the hyperbolic volume and the Chern--Simons invariant of the complements of the knots. As higher coefficients of the expansions, we obtain a new series of invariants of these knots.A non-trivial part of the proof is to apply the saddle point method to calculate the asymptotic expansion of an integral which presents the Kashaev invariant. A key step of this part is to give a concrete homotopy of the (real 3-dimensional) domain of the integral in ℂ3 in such a way that the boundary of the domain always stays in a certain domain in ℂ3 given by the potential function of the hyperbolic structure.

On the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings

2017 ◽  
Vol 28 (13) ◽  
pp. 1750096 ◽  
Author(s):  
Tomotada Ohtsuki
Keyword(s):  
Asymptotic Expansion ◽  
Saddle Point ◽  
Asymptotic Expansions ◽  
Technical Difficulty ◽  
Point Method ◽  
Saddle Point Method ◽  
Cambridge Philos ◽  
Volume Conjecture ◽  
Hyperbolic Volume ◽  
Chern Simons

We give presentations of the asymptotic expansions of the Kashaev invariant of hyperbolic knots with seven crossings. As the volume conjecture states, the leading terms of the expansions present the hyperbolic volume and the Chern–Simons invariant of the complements of the knots. As coefficients of the expansions, we obtain a series of new invariants of the knots. This paper is a continuation of the previous papers [T. Ohtsuki, On the asymptotic expansion of the Kashaev invariant of the [Formula: see text] knot, Quantum Topol. 7 (2016) 669–735; T. Ohtsuki and Y. Yokota, On the asymptotic expansion of the Kashaev invariant of the knots with 6 crossings, to appear in Math. Proc. Cambridge Philos. Soc.], where the asymptotic expansions of the Kashaev invariant are calculated for hyperbolic knots with five and six crossings. A technical difficulty of this paper is to use 4-variable saddle point method, whose concrete calculations are far more complicated than the previous papers.

The Transition Matrix Between the Specht and 𝔰𝔩3 Web Bases is Unitriangular With Respect to Shadow Containment

2020 ◽  
Author(s):  
Heather M Russell ◽  
Julianna Tymoczko
Keyword(s):  
Symmetric Group ◽  
Partial Order ◽  
Transition Matrix ◽  
Group Representation ◽  
Jones Polynomial ◽  
Basis Matrix ◽  
Combinatorial Structures ◽  
Quantum Invariants ◽  
One Dimensional ◽  
Link Diagrams

Abstract Webs are planar graphs with boundary that describe morphisms in a diagrammatic representation category for $\mathfrak{sl}_k$. They are studied extensively by knot theorists because braiding maps provide a categorical way to express link diagrams in terms of webs, producing quantum invariants like the well-known Jones polynomial. One important question in representation theory is to identify the relationships between different bases; coefficients in the change-of-basis matrix often describe combinatorial, algebraic, or geometric quantities (e.g., Kazhdan–Lusztig polynomials). By ”flattening” the braiding maps, webs can also be viewed as the basis elements of a symmetric group representation. In this paper, we define two new combinatorial structures for webs: band diagrams and their one-dimensional projections, shadows, which measure depths of regions inside the web. As an application, we resolve an open conjecture that the change of basis between the so-called Specht basis and web basis of this symmetric group representation is unitriangular for $\mathfrak{sl}_3$-webs ([ 33] and [ 29].) We do this using band diagrams and shadows to construct a new partial order on webs that is a refinement of the usual partial order. In fact, we prove that for $\mathfrak{sl}_2$-webs, our new partial order coincides with the tableau partial order on webs studied by the authors and others [ 12, 17, 29, 33]. We also prove that though the new partial order for $\mathfrak{sl}_3$-webs is a refinement of the previously studied tableau order, the two partial orders do not agree for $\mathfrak{sl}_3$.

THE COLORED JONES POLYNOMIALS OF DOUBLES OF KNOTS

2008 ◽  
Vol 17 (08) ◽  
pp. 925-937
Author(s):  
TOSHIFUMI TANAKA
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Volume Conjecture ◽  
Jones Polynomials ◽  
Skein Theory ◽  
Colored Jones Polynomials

We give formulas for the N-colored Jones polynomials of doubles of knots by using skein theory. As a corollary, we show that if the volume conjecture for untwisted positive (or negative) doubles of knots is true, then the colored Jones polynomial detects the unknot.

scholarly journals COLORED JONES POLYNOMIALS WITH POLYNOMIAL GROWTH

2008 ◽  
Vol 10 (supp01) ◽  
pp. 815-834 ◽  
Author(s):  
KAZUHIRO HIKAMI ◽  
HITOSHI MURAKAMI
Keyword(s):  
Irreducible Representation ◽  
Complex Number ◽  
Polynomial Growth ◽  
Jones Polynomial ◽  
The Other ◽  
Absolute Value ◽  
Volume Conjecture ◽  
Jones Polynomials ◽  
Dimensional Irreducible Representation ◽  
Colored Jones Polynomials

The volume conjecture and its generalizations say that the colored Jones polynomial corresponding to the N-dimensional irreducible representation of sl(2;ℂ) of a (hyperbolic) knot evaluated at exp(c/N) grows exponentially with respect to N if one fixes a complex number c near [Formula: see text]. On the other hand if the absolute value of c is small enough, it converges to the inverse of the Alexander polynomial evaluated at exp c. In this paper we study cases where it grows polynomially.

scholarly journals Combinatorics of link diagrams and volume

2015 ◽  
Vol 24 (01) ◽  
pp. 1550001 ◽  
Author(s):  
Adam Giambrone
Keyword(s):  
Jones Polynomial ◽  
Colored Jones Polynomial ◽  
Rich Family ◽  
Link Diagrams ◽  
Volume Conjecture ◽  
Twist Number

We show that the volumes of certain hyperbolic A-adequate links can be bounded (above and) below in terms of two diagrammatic quantities: the twist number and the number of certain alternating tangles in an A-adequate diagram. We then restrict our attention to plat closures of certain braids, a rich family of links whose volumes can be bounded in terms of the twist number alone. Furthermore, in the absence of special tangles, our volume bounds can be expressed in terms of a single stable coefficient of the colored Jones polynomial. Consequently, we are able to provide a new collection of links that satisfy a Coarse Volume Conjecture.

scholarly journals Disentangling a deep learned volume formula

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Jessica Craven ◽  
Vishnu Jejjala ◽  
Arjun Kar
Keyword(s):  
Jones Polynomial ◽  
Simons Theory ◽  
Average Error ◽  
Approximation Formula ◽  
Expectation Values ◽  
Reverse Engineer ◽  
Hyperbolic Volume ◽  
Integration Cycle ◽  
Chern Simons ◽  
Large Improvement

Abstract We present a simple phenomenological formula which approximates the hyperbolic volume of a knot using only a single evaluation of its Jones polynomial at a root of unity. The average error is just 2.86% on the first 1.7 million knots, which represents a large improvement over previous formulas of this kind. To find the approximation formula, we use layer-wise relevance propagation to reverse engineer a black box neural network which achieves a similar average error for the same approximation task when trained on 10% of the total dataset. The particular roots of unity which appear in our analysis cannot be written as e2πi/(k+2) with integer k; therefore, the relevant Jones polynomial evaluations are not given by unknot-normalized expectation values of Wilson loop operators in conventional SU(2) Chern-Simons theory with level k. Instead, they correspond to an analytic continuation of such expectation values to fractional level. We briefly review the continuation procedure and comment on the presence of certain Lefschetz thimbles, to which our approximation formula is sensitive, in the analytically continued Chern-Simons integration cycle.

Sign in / Sign up

Export Citation Format

Share Document