A diagrammatic approach to the AJ Conjecture
- PDF / 846,908 Bytes
- 38 Pages / 439.37 x 666.142 pts Page_size
- 47 Downloads / 227 Views
Mathematische Annalen
A diagrammatic approach to the AJ Conjecture Renaud Detcherry1 · Stavros Garoufalidis2 Received: 11 July 2019 / Revised: 29 May 2020 © The Author(s) 2020
Abstract The AJ Conjecture relates a quantum invariant, a minimal order recursion for the colored Jones polynomial of a knot (known as the Aˆ polynomial), with a classical invariant, namely the defining polynomial A of the PSL2 (C) character variety of a knot. More precisely, the AJ Conjecture asserts that the set of irreducible factors ˆ of the A-polynomial (after we set q = 1, and excluding those of L-degree zero) coincides with those of the A-polynomial. In this paper, we introduce a version of the ˆ A-polynomial that depends on a planar diagram of a knot (that conjecturally agrees ˆ with the A-polynomial) and we prove that it satisfies one direction of the AJ Conjecture. Our proof uses the octahedral decomposition of a knot complement obtained from a planar projection of a knot, the R-matrix state sum formula for the colored Jones polynomial, and its certificate. Mathematics Subject Classification Primary 57N10; Secondary 57M25
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The colored Jones polynomial and the AJ Conjecture 1.2 q-holonomic sums . . . . . . . . . . . . . . . . . . . 1.3 Our results . . . . . . . . . . . . . . . . . . . . . . . 1.4 Sketch of the proof . . . . . . . . . . . . . . . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
Communicated by Thomas Schick.
B
Stavros Garoufalidis [email protected] http://people.mpim-bonn.mpg.de/stavros Renaud Detcherry [email protected] http://people.mpim-bonn.mpg.de/detcherry
1
Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
2
International Center for Mathematics, Department of Mathematics, Southern University of Science and Technology, Shenzhen, China
123
R. Detcherry, S. Garoufalidis 2 Knot diagrams, their octahedral decomposition and their gluing equations . . . . 2.1 Ideal triangulations and their gluing equations . . . . . . . . . . . . . . . . 2.2 Spines and gluing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The octahedral decomposition of a knot diagram . . . . . . . . . . . . . . . 2.4 The spine of the 5T -triangulation of a knot diagram and its gluing equations 2.5 Labeled knot diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Analysis of triangle and shingle relations . . . . . . . . . . . . . . . . . . . 2.7 Analysis of big region equations . . . . . . . . . . . . . . . . . . . . . . . 2.8 Formulas for the loop equations . . . . . . . . . . . . . . . . . . . . . . . . 2.9 A square root of the holonomy of the longitude . . . . . . . . . . . . . . . 3 q-holonomic functions, creative telescoping and certificates . . . . . . . . . . . 4 The colored Jone
Data Loading...
