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European Mathematical Society Publishing House
2024-03-29 12:10:59
5
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=QT&vol=9&iss=3&update_since=2024-03-29
Quantum Topology
Quantum Topol.
QT
1663-487X
1664-073X
General
10.4171/QT
http://www.ems-ph.org/doi/10.4171/QT
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European Mathematical Society Publishing House
Zuerich, Switzerland
© European Mathematical Society
9
2018
3
Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants
Qingtao
Chen
ETH Zürich, Switzerland
Tian
Yang
Stanford University, USA
Reshetikhin–Turaev invariants, Turaev–Viro invariants, volume conjecture
We consider the asymptotics of the Turaev–Viro and the Reshetikhin–Turaev invariants of a hyperbolic 3-manifold, evaluated at the root of unity exp$({2\pi\sqrt{-1}}/{r})$ instead of the standard exp$({\pi\sqrt{-1}}/{r})$. We present evidence that, as $r$ tends to $\infty$, these invariants grow exponentially with growth rates respectively given by the hyperbolic and the complex volume of the manifold. This reveals an asymptotic behavior that is different from that of Witten's Asymptotic Expansion Conjecture, which predicts polynomial growth of these invariants when evaluated at the standard root of unity. This new phenomenon suggests that the Reshetikhin–Turaev invariants may have a geometric interpretation other than the original one via $SU(2)$ Chern–Simons gauge theory.
Manifolds and cell complexes
Quantum theory
419
460
10.4171/QT/111
http://www.ems-ph.org/doi/10.4171/QT/111
7
9
2018
The Homflypt polynomial and the oriented Thompson group
Valeriano
Aiello
Vanderbilt University, Nashville, USA
Roberto
Conti
Università di Roma La Sapienza, Italy
Vaughan
Jones
Vanderbilt University, Nashville, USA
Unitary representations, positive definite functions, Thompson group F , binary trees, category of forests, group of fractions, HOMFLYPT polynomial, tangles, knots, oriented link invariants, TQFT, subfactors, planar algebras, CFT
We show how to construct unitary representations of the oriented Thompson group $\vec F $ from oriented link invariants. In particular we show that the suitably normalised HOMFLYPT polynomial defines a positive definite function of $\vec F$.
Manifolds and cell complexes
Group theory and generalizations
Abstract harmonic analysis
461
472
10.4171/QT/112
http://www.ems-ph.org/doi/10.4171/QT/112
7
9
2018
The classification of $3^n$ subfactors and related fusion categories
Masaki
Izumi
Kyoto University, Japan
Subfactors, fusion categories, Cuntz algebras
We investigate a (potentially infinite) series of subfactors, called $3^n$ subfactors, including $A_4$, $A_7$, and the Haagerup subfactor as the first three members corresponding to $n=1,2,3$. Generalizing our previous work for odd $n$, we further develop a Cuntz algebra method to construct $3^n$ subfactors and show that the classification of the $3^n$ subfactors and related fusion categories is reduced to explicit polynomial equations under a mild assumption, which automatically holds for odd $n$. In particular, our method with $n=4$ gives a uniform construction of $4$ finite depth subfactors, up to dual, without intermediate subfactors of index $3+\sqrt{5}$. It also provides a key step for a new construction of the Asaeda–Haagerup subfactor due to Grossman, Snyder, and the author.
Functional analysis
Category theory; homological algebra
473
562
10.4171/QT/113
http://www.ems-ph.org/doi/10.4171/QT/113
7
9
2018
The Khovanov homology of infinite braids
Gabriel
Islambouli
University of Virginia, Charlottesville, USA
Michael
Willis
University of Virginia, Charlottesville, USA
Khovanov homology, Khovanov homotopy type, colored Khovanov homology, Jones–Wenzl projectors, infinite braids
We show that the limiting Khovanov chain complex of any infinite positive braid categorifies the Jones–Wenzl projector. This result extends Lev Rozansky’s categorification of the Jones–Wenzl projectors using the limiting complex of infinite torus braids. We also show a similar result for the limiting Lipshitz–Sarkar–Khovanov stable homotopy types of the closures of such braids. Extensions to more general infinite braids are also considered.
Manifolds and cell complexes
Algebraic topology
563
590
10.4171/QT/114
http://www.ems-ph.org/doi/10.4171/QT/114
7
9
2018
Triangular decomposition of skein algebras
Thang T. Q.
Lê
Georgia Institute of Technology, Atlanta, USA
Kauffman bracket skein module, quantum trace map, ideal triangulation
By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The newskein algebra of an ideal triangle has a simple presentation. This gives an easy proof of the existence of the quantum trace map of Bonahon and Wong. We also explain the relation between our skein algebra and the one defined by Muller, and use it to show that the quantum trace map can be extended to the Muller skein algebra.
Manifolds and cell complexes
General
591
632
10.4171/QT/115
http://www.ems-ph.org/doi/10.4171/QT/115
7
9
2018