Quantum Approximation of the Jones Polynomial, 2005; Aharonov, Jones, Landau

2011 ◽  
Keyword(s):  
Jones Polynomial ◽  
Quantum Approximation

Quantum Approximation of the Jones Polynomial

2008 ◽  
pp. 700-702
Author(s):  
Zeph Landau
Keyword(s):  
Jones Polynomial ◽  
Quantum Approximation

Quantum Approximation of the Jones Polynomial

2016 ◽  
pp. 1691-1695
Author(s):  
Zeph Landau
Keyword(s):  
Jones Polynomial ◽  
Quantum Approximation

Quantum Approximation of the Jones Polynomial

2015 ◽  
pp. 1-6
Author(s):  
Zeph Landau
Keyword(s):  
Jones Polynomial ◽  
Quantum Approximation

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 $\mathfrak{sl}_3$.

An evaluation of the Jones polynomial of a parallel link

1988 ◽  
Vol 104 (1) ◽  
pp. 105-113
Author(s):  
Makoto Sakuma
Keyword(s):  
Jones Polynomial ◽  
Branched Covering ◽  
Parallel Link ◽  
Image Position

The Jones polynomial VL(t) of a link L in S3 contains certain information on the homology of the 2-fold branched covering D(L) of S3 branched along L. The following formulae are proved by Jones[3] and Lickorish and Millett[6] respectively:

Jones polynomial invariants for knots and satellites

1991 ◽  
Vol 109 (1) ◽  
pp. 83-103 ◽  
Author(s):  
H. R. Morton ◽  
P. Strickland
Keyword(s):  
Quantum Group ◽  
Linear Combination ◽  
Simple Formula ◽  
Jones Polynomial ◽  
Polynomial Invariants ◽  
Satellite Link ◽  
Representation Ring ◽  
Parallel Links ◽  
Jones Polynomials ◽  
Special Case

AbstractResults of Kirillov and Reshetikhin on constructing invariants of framed links from the quantum group SU(2)q are adapted to give a simple formula relating the invariants for a satellite link to those of the companion and pattern links used in its construction. The special case of parallel links is treated first. It is shown as a consequence that any SU(2)q-invariant of a link L is a linear combination of Jones polynomials of parallels of L, where the combination is determined explicitly from the representation ring of SU(2). As a simple illustration Yamada's relation between the Jones polynomial of the 2-parallel of L and an evaluation of Kauffman's polynomial for sublinks of L is deduced.

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 The Jones polynomial of periodic knots

1991 ◽  
Vol 113 (3) ◽  
pp. 889-889 ◽  
Author(s):  
Yoshiyuki Yokota
Keyword(s):  
Jones Polynomial

scholarly journals The Jones polynomial and graphs on surfaces

2008 ◽  
Vol 98 (2) ◽  
pp. 384-399 ◽  
Author(s):  
Oliver T. Dasbach ◽  
David Futer ◽  
Efstratia Kalfagianni ◽  
Xiao-Song Lin ◽  
Neal W. Stoltzfus
Keyword(s):  
Jones Polynomial ◽  
Graphs On Surfaces
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