April 2018
·
1 Read
Let Sigma be a closed oriented surface of genus g. We show that the Kauffman bracket skein module of Sigma x S^1 over the field of rational functions in A has dimension at least 2^{2g+1}+2g-1.
What is this page?
This page lists works of an author who doesn't have a ResearchGate profile or hasn't added the works to their profile yet. It is automatically generated from public (personal) data to further our legitimate goal of comprehensive and accurate scientific recordkeeping. If you are this author and want this page removed, please let us know.
Publications (47)
April 2018
·
27 Reads
·
15 Citations
Proceedings of the American Mathematical Society
Let Sigma be a closed oriented surface of genus g. We show that the Kauffman bracket skein module of Sigma x S^1 over the field of rational functions in A has dimension at least 2^{2g+1}+2g-1.
![Lollipop tree Gg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_g$$\end{document}](https://www.researchgate.net/publication/304642571/figure/fig3/AS:960038076182530@1605902322635/Lollipop-tree-Ggdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
![The colorings associated to the highest weights in Case II\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {II}$$\end{document}, Case III\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {III}$$\end{document}, and Case IV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {IV}$$\end{document}. The leftmost part of Gg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G_g$$\end{document} is not drawn as it is colored zero. For the same reason the rightmost edge is not drawn in Case IV\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {IV}$$\end{document} as c is zero. These graphs could be guessed by taking the smallest coloring which is rightmost on the graph and has the given type](https://www.researchgate.net/publication/304642571/figure/fig4/AS:960038076182539@1605902322948/The-colorings-associated-to-the-highest-weights-in-Case-IIdocumentclass12ptminimal_Q320.jpg)
November 2017
·
26 Reads
·
7 Citations
Inventiones mathematicae
For p≥5 a prime, and g≥3 an integer, we use Topological Quantum Field Theory (TQFT) to study a family of p-1 highest weight modules Lp(λ) for the symplectic group Sp(2g,K) where K is an algebraically closed field of characteristic p. This permits explicit formulae for the dimension and the formal character of Lp(λ) for these highest weights.
August 2016
·
8 Reads
·
2 Citations
Glasgow Mathematical Journal
We use elementary skein theory to prove a version of a result of Stylianakis who showed that under mild restrictions on m and n, the normal closure of the m-th power of a half-twist has infinite index in the mapping class group of a sphere with 2n punctures.
August 2016
We use elementary skein theory to prove a version of a result of Stylianakis who showed that under mild restrictions on m and n, the normal closure of the m-th power of a half-twist has infinite index in the mapping class group of a sphere with 2n punctures.
June 2016
For p>3 a prime, and g>2 an integer, we use Topological Quantum Field Theory (TQFT) to study a family of p-1 highest weight modules L_p(lambda) for the symplectic group Sp(2g,K) where K is an algebraically closed field of characteristic p. This permits explicit formulae for the dimension and the formal character of L_p(lambda) for these highest weights.
May 2016
·
12 Reads
Proceedings of the Steklov Institute of Mathematics
Let I" (g,b) denote the orientation-preserving mapping class group of a closed orientable surface of genus g with b punctures. For a group G let I broken vertical bar (f) (G) denote the intersection of all maximal subgroups of finite index in G. Motivated by a question of Ivanov as to whether I broken vertical bar (f) (G) is nilpotent when G is a finitely generated subgroup of I" (g,b) , in this paper we compute I broken vertical bar (f) (G) for certain subgroups of I" (g,b) . In particular, we answer Ivanov's question in the affirmative for these subgroups of I"g,b.
January 2015
·
8 Reads
Oberwolfach Reports
This workshop brought together specialists in areas ranging from arithmetic groups to topological quantum field theory, with common interest in arithmetic aspects of discrete groups arising from topology. The meeting showed significant progress in the field and enhanced the many connections between its subbranches.
December 2014
·
28 Reads
·
1 Citation
Труды Математического института им Стеклова
Let Gamma_{g,b} denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus g with b punctures. For a group G let Phi_f(G) denote the intersection of all maximal subgroups of finite index in G. In this paper we prove several results about Phi_f(G) for certain subgroups of Gamma_{g,b}. In particular, this answers a question of Ivanov for these subgroups of Gamma_{g,b}.
November 2011
·
42 Reads
·
5 Citations
Journal of Pure and Applied Algebra
We compare the dimensions of the irreducible Sp(2g,K)-modules over a field K of characteristic p constructed by Gow with the dimensions of the irreducible Sp(2g,F_p)-modules that appear in the first approximation to representations of mapping class groups of surfaces in Integral Topological Quantum Field Theory. For this purpose, we derive a trigonometric formula for the dimensions of Gow's representations. This formula is equivalent to a special case of a formula contained in unpublished work of Foulle. Our direct proof is simpler than the proof of Foulle's more general result, and is modeled on the proof of the Verlinde formula in TQFT.
Citations (36)
... More deep studies in this vein were carried out in [8,9]. Computational results were seen in [1,4,[10][11][12][15][16][17] and the references therein. However, skein modules still appear rather mysterious; there remain many open problems about structures, properties of skein modules, as well as connections to classical invariants. ...
- Citing Article
April 2018
Proceedings of the American Mathematical Society
... More precisely, those invariants are elements of the ring of integers of Q(ζ), namely, Z[ζ]. The result was reproved in [MR97], generalized to all classical Lie types in [MW98,TY99], then to all Lie types by Le [Le03]. These results helped us relate the quantum invariants to other invariants such as the Casson invariant [Mur94,Mur95] and the Ohtsuki series [Oht95,Oht96,Le03]. ...
- Citing Article
December 1998
... where P m (µ i jk (L)) means the result of evaluating the polynomial P m at y i jk = µ i jk (L). For m = 3, we find the Cochran's formula (31), and for m ≥ 5 the formula is from [56], which obtains that the first non-vanishing coefficient of ∇ L (z) for algebraically split links with 5 components is equal to: where P 5 (µ i jk (L)) consists of 15 terms corresponding to the spanning trees of Γ 5 . ...
- Citing Conference Paper
September 2002
Geometry and Topology Monographs
... The trivalent graph notation derives many useful formulae of the Kauffman bracket and makes it easy for us to calculate the quantum representations. Masbaum [Mas17] proved that the normal closure of powers of half-twists has infinite index in mapping class groups of punctured spheres by the quantum representation of braid groups. He used elementary skein theory whereas Stylianakis [Sty15] used the Jones representation [Jon87]. ...
- Citing Article
August 2016
Glasgow Mathematical Journal
... For example, certain families of quantum representations are known to be asymptotically linear, i.e., the intersection of the kernel of all representations in a family is trivial ( [And06], [FWW02]). Many quantum representations are also known to be integral ( [Gil04]), and they can be used to study the modular representation theory of mapping class groups and symplectic groups ( [GM17]). ...
- Citing Article
- Publisher preview available
November 2017
Inventiones mathematicae
... Thus H 1 g and Γ 1 g are residually simple. We derive that the Frattini subgroup is trivial for Γ 1 g , complementing the result obtained in [51] for closed surfaces: ...
- Citing Article
- Full-text available
December 2014
Труды Математического института им Стеклова
... After that, Korinman [12] studied non-prime case: He showed that the representations are irreducible when p is the product of two distinct odd primes and when p is square of an odd prime, and for some other p, decomposable examples have been given. In [7], they first studied the cases that are with boundary. They proved when p is an odd prime, the representations of any surfaces with one boundary component is irreducible. ...
- Citing Article
- Full-text available
October 2011
Pacific Journal of Mathematics
... The natural inclusion i: S 1 × D 2 → S 1 × S 2 induces an epimorphism i * : K(S 1 × D 2 ) → K(S 1 × S 2 ). Following Masbaum's work in the case of the Kauffman bracket skein module [5], we will use the following method to parametrize the relations arising from sliding over the 2-handle. Pick two points A, B on γ, which decompose γ into two intervals γ ′ and γ ′′ . ...
Reference:
On the Kauffman skein modules
- Citing Article
December 1996
manuscripta mathematica
... Remark that the TQFT described in Theorem 6.5 is different from those studied, for instance, in [34,5] in that the target category is not that of vector spaces. Rather, when working over a field, it seems to fit very well in the general framework of [4] where in particular a cp-rigid and cocomplete braided monoidal category is shown to be a 3dualisable object in the 4-category BrTens so that, as a consequence, there is an extended TQFT associated to it. ...
- Citing Article
October 1992
Topology
... He defined this invariant in terms of the Feynman path integral. Rigorous definitions of the invariant for SU(2) have been given by[19,10,3,8]. Specially, in[10], Lickorish redefined the invariant introducing the linear skein theory and the Temperley–Lieb algebra. ...
- Citing Article