scholarly journals Quantum invariants of periodic three-manifolds

1999 ◽  
Author(s):  
Patrick M Gilmer
Keyword(s):  
Quantum Invariants

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 $\mathfrak{sl}_3$.

Quantum Invariants of 3-Manifolds

2001 ◽  
pp. 707-734 ◽  
Author(s):  
W RAYMONDLICKORISH
Keyword(s):  
Quantum Invariants

scholarly journals Quantum invariants of random 3–manifolds

2011 ◽  
Vol 11 (4) ◽  
pp. 2191-2205 ◽  
Author(s):  
Nathan M Dunfield ◽  
Helen Wong
Keyword(s):  
Quantum Invariants

scholarly journals SO(3) quantum invariants are dense

2009 ◽  
Vol 148 (2) ◽  
pp. 289-295 ◽  
Author(s):  
HELEN WONG
Keyword(s):  
Complex Plane ◽  
Quantum Invariants ◽  
Dense Set

AbstractWe show that whenr≥ 5 is prime, the SO(3) Witten–Reshetikhin–Turaev quantum invariants for three-manifolds at the levelrform a dense set in the complex plane. This confirms a conjecture of Larsen and Wang.

scholarly journals Homology TQFT's and the Alexander–Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory

2003 ◽  
Vol 55 (4) ◽  
pp. 766-821 ◽  
Author(s):  
Thomas Kerler
Keyword(s):  
Moduli Spaces ◽  
Hopf Algebras ◽  
Large Family ◽  
Quantum Invariants ◽  
Topological Quantum ◽  
Irreducible Components ◽  
Skein Theory ◽  
Higher Rank ◽  
The One ◽  
Representation Varieties

AbstractWe develop an explicit skein-theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the homology ofU(1)-representation varieties on the one side and the combinatorially constructed Hennings TQFT based on the quasitriangular Hopf algebra= ℤ/2 n ⋊ Λ* ℝ2on the other side. We find that both TQFT's are SL(2; ℝ)-equivariant functors and, as such, are isomorphic. The SL(2; ℝ)-action in the Hennings construction comes from the natural action onand in the case of the Frohman–Nicas theory from the Hard–Lefschetz decomposition of theU(1)-moduli spaces given that they are naturally Kähler. The irreducible components of this TQFT, corresponding to simple representations of SL(2; ℤ) and Sp(2g; ℤ), thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg–Witten theories, Casson type theories for homology circlesà laDonaldson, higher rank gauge theories following Frohman and Nicas, and the ℤ=pℤ reductions of Reshetikhin.Turaev theories over the cyclotomic integers ℤ[ζp]. We also conjecture that the Hennings TQFT for quantum-sl2is the product of the Reshetikhin–Turaev TQFT and such a homological TQFT.

scholarly journals Wall-crossing in genus zero Landau–Ginzburg theory

2017 ◽  
Vol 2017 (733) ◽  
Author(s):  
Dustin Ross ◽  
Yongbin Ruan
Keyword(s):  
Moduli Spaces ◽  
Homogeneous Polynomial ◽  
Genus Zero ◽  
Quantum Invariants ◽  
Rational Parameter ◽  
The Family ◽  
Wall Crossing

AbstractWe study a family of moduli spaces and corresponding quantum invariants introduced recently by Fan–Jarvis–Ruan. The family has a wall-and-chamber structure relative to a positive rational parameter ϵ. For a Fermat quasi-homogeneous polynomial

Non Semi-Simple ${\mathfrak {sl}(2)}$ Quantum Invariants, Spin Case

2014 ◽  
Vol 39 (4) ◽  
pp. 481-495
Author(s):  
Christian Blanchet ◽  
Francesco Costantino ◽  
Nathan Geer ◽  
Bertrand Patureau-Mirand
Keyword(s):  
Quantum Invariants
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