DOCSLIB.ORG

Education 2015 Michigan State University, East Lansing

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Education 2015 Michigan State University, East Lansing CURRICULUM VITAE Christine Ruey Shan Lee Department of mathematics, Email: [email protected] The University of Texas at Austin, Website: www.ma.utexas.edu/users/clee 2515 Speedway, RLM 8.100, Austin, TX 78712 Education 2015 Michigan State University, East Lansing, MI, U.S.A. Ph.D. in Mathematics, May 2015 Advisor: Efstratia Kalfagianni Dissertation title: Jones-type link invariants and applications to 3-manifold topology. 2008 The University of Manchester, Manchester, UK M.Sc. with distinction in Pure Mathematics, September 2008 2007 California Institute of Technology, Pasadena, CA, U.S.A. B.S. in Mathematics, June 2007 Academic Employment 2015- NSF postdoctoral fellow and instructor, University of Texas at present Austin, Austin, TX. 2017 Fall Visiting scholar, the Max Planck Institute for Mathematics, Bonn, Germany. 2009-15 Teaching and research assistant, Michigan State University, East Lansing, MI. Research Interests: Knot theory; quantum topology; 3-manifolds Publications 7. Jones slopes and coarse volume of near-alternating links, available at arXiv:1708.04900. 6. Normal and Jones surfaces of knots, with Efstratia Kalfagianni, available 1 at arXiv:1702.06466, submitted. 5. A trivial tail homology for non A-adequate links, accepted for publication in Algebraic and Geometric topology, available at arXiv:1611.00686. 4. Slopes for pretzel knots, with Roland van der Veen, New York Journal of Mathematics, 22, p. 1339{1364, 2016. 3. Stability properties of the colored Jones polynomial, available at arXiv:1409.4457, submitted. 2. Crosscap numbers and the Jones polynomial, with Efstratia Kalfagianni, Advances in Mathematics, 286, p.308-337, 2016. 1. On the degree of the colored Jones polynomial, with Efstratia Kalfagianni, Acta Mathematica Vietnamica (Proceedings of Quantum Topology and Hyperbolic Geometry in Nha Trang, May 2013), 39(4), 2014. Preprints • Tangle replacements and Jones slopes for Montesinos knots, with Stavros Garoufalidis and Roland van der Veen, in preparation. • Some computations on the transverse invariant from Khovanov homology, with Diana Hubbard, available at https://www.ma.utexas.edu/users/clee/ TransverseComp.pdf • On 3-braids and L-space knots, with Faramarz Vafaee, available at https: //www.ma.utexas.edu/users/clee/3braidpaper.pdf Awards and Fellowships 2015-18 NSF Postdoctoral Fellowship MSPRF-DMS 1502860. 2014 Michigan State University College of Natural Science Disser- tation Completion Fellowship. 2009-11 Geometry/Topology Research Training Group Fellowship, sup- ported by an NSF/RTG grant. Recent Invited and Conference Talks 2 • Stability properties of the colored Jones polynomial, Nov 9th, 2017, Al- gebra and Number Theory Seminar, University College Dublin, Dublin, Ireland. • The colored Jones polynomial and surfaces in 3-manifolds, Sep 28th, 2017, S´eminaireTopologies, Institut Montpelli´erainAlexander Grothendieck. • The colored Jones polynomial and surfaces in 3-manifolds, Aug 21st, 2017, Oberwolfach Workshop in Low-dimensional Topology and Number Theory, Oberwolfach, Germany. • Jones slopes and coarse volume for near-alternating links, May 6th, 2017, Special session on Invariants in Low-dimensional Topology, Spring Eastern Sectional Meeting, New York, NY. • Jones slopes and Murasugi sums of links, March 22nd, 2017, Virtual Seminar, Louisiana State University, Baton Rouge, LA. • Jones slopes and Murasugi sums of links, February 20th, 2017, Geometry and Topology Seminar, Georgia Institute of Technology, Atlanta, CA. • Jones slopes and Murasugi sums of links, February 9th, 2017, Topology Seminar, Michigan State University, East Lansing, MI. • Slopes for near-adequate knots, January 6th, 2017, Special session on Women in Topology, Joint Mathematics Meeting, Atlanta, GA. • A trivial tail homology for non A-adequate links, December 4th, 2016, Session on Geometric Group Theory and Topology in Low Dimensions, CMS Winter Meeting, Niagara Falls, ON, Canada. • Stability in the categorification of the colored Jones polynomial, October 9th, 2016, Special session on Floer theoretic invariants of 3-manifolds and knots, University of Denver, Denver, CO. • Quantum knot invariants and 3-manifolds, August 9th, 2016, Workshop on knot theory and related topics, RIMS, Kyoto, Japan. • The colored Jones polynomial and boundary slopes of pretzel knots, July 19th, 2016, Knots in Hellas, Ancient Olympia, Greece. • Slope conjectures for pretzel knots, April 20th, 2016, Topology Seminar, California Institute of Technology, Pasadena, CA. 3 Teaching Experience 2017 Instructor, University of Texas at Austin, M367K: Topology I. Spring 2011-15 Teaching Assistant, Michigan State University MTH 869: Geometry and Topology II { Recitation instructor MTH 133: Calculus II { Recitation instructor MTH 234: Multivariable Calculus { Recitation instructor, Pri- mary instructor MTH 132: Calculus I { Primary instructor MTH 124: Survey of Calculus I { Primary instructor Service & Outreach 2018 January 11-12th: Co-organizer of the special session \Quan- tum link invariants, Khovanov homology, and low-dimensional manifolds" at the AMS Joint Mathematics Meeting, San Diego, CA. 2017 April 5th #NotSoMath: The pigeon in the hole, a general Spring audience talk on the pigeonhole principle. #NotSoMath is a series of events held every other month at the public venue the North Door in Austin. An event consists of 2 or 3 talks, each of about 20-30 minutes by mathematicians aimed at the general public. Here is a current link to the latest event. http://do512.com/events/2017/10/4/not-so-math, and a link to the math books on sale at the event I presented. http://library.austintexas.gov/event/notsomath-386771 Sample titles of talks: The mathematics of gerrymandering by Anton Dochertmann (Texas State), Cryptography and Security by Lubjuana Behsaj (UT), Mathsplaining by Pedro Fernando Morales-Almazan (UT). 4 2017 Participant of the Quantum Topology and Categorification Spring Seminar of the Math department, University of Texas at Austin. 2016-17 A Mentor of the Directed Reading Program (DRP), University Spring of Texas at Austin. Student: Helen Smith Title of presentation: The Feynman path integral. 2015 Graduate student volunteer for the Middle School Girls Math Spring & Science day, Michigan State University 2014-15 Co-organizer of the Graduate Student Geometry/Topology Seminar of the Math department, Michigan State University 2013-15 Co-founder and Organizer of the Math Department Tea Time, Michigan State University Referee Transactions of the AMS Proceedings of the AMS The New York Journal of Mathematics Skills • Programming: C++, Python, Mathematica • Language: I am fluent in English and Mandarin and proficient in French. 5.
Load more

Recommended publications
  • Finite Type Invariants for Knots 3-Manifolds
    Pergamon Topology Vol. 37, No. 3. PP. 673-707, 1998 ~2 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0040-9383/97519.00 + 0.00 PII: soo4o-9383(97)00034-7 FINITE TYPE INVARIANTS FOR KNOTS IN 3-MANIFOLDS EFSTRATIA KALFAGIANNI + (Received 5 November 1993; in revised form 4 October 1995; final version 16 February 1997) We use the Jaco-Shalen and Johannson theory of the characteristic submanifold and the Torus theorem (Gabai, Casson-Jungreis)_ to develop an intrinsic finite tvne__ theory for knots in irreducible 3-manifolds. We also establish a relation between finite type knot invariants in 3-manifolds and these in R3. As an application we obtain the existence of non-trivial finite type invariants for knots in irreducible 3-manifolds. 0 1997 Elsevier Science Ltd. All rights reserved 0. INTRODUCTION The theory of quantum groups gives a systematic way of producing families of polynomial invariants, for knots and links in [w3 or S3 (see for example [18,24]). In particular, the Jones polynomial [12] and its generalizations [6,13], can be obtained that way. All these Jones-type invariants are defined as state models on a knot diagram or as traces of a braid group representation. On the other hand Vassiliev [25,26], introduced vast families of numerical knot invariants (Jinite type invariants), by studying the topology of the space of knots in [w3. The compu- tation of these invariants, involves in an essential way the computation of related invariants for special knotted graphs (singular knots). It is known [l-3], that after a suitable change of variable the coefficients of the power series expansions of the Jones-type invariants, are of Jinite type.
    [Show full text]
  • Meetings & Conferences of The
    Meetings & Conferences of the AMS IMPORTANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found on the AMS website. See http://www.ams.org/meetings/. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL and in an electronic issue of the Notices as noted below for each meeting. sity, Youssef Raffoul, University of Dayton, and Agacik St. Louis, Missouri Zafer, American University of the Middle East. Advances in Mathematical Methods for Disease Model- Washington University ing, Jimin Ding, Washington University in St. Louis, Necibe October 18–20, 2013 Tuncer, University of Tulsa, and Naveen K. Vaidya, Uni- versity of Missouri-Kansas City. Friday – Sunday Algebraic Cycles and Coherent Sheaves, Roya Beheshti, Meeting #1094 Matt Kerr, and N. Mohan Kumar, Washington University in St. Louis. Central Section Associate secretary: Georgia M. Benkart Algebraic and Combinatorial Invariants of Knots, Announcement issue of Notices: August 2013 Heather Dye, McKendree University, Allison Henrich, Program first available on AMS website: September 5, 2013 Seattle University, Aaron Kaestner, North Park University, Program issue of electronic Notices: October 2013 and Louis Kauffman, University of Illinois. Issue of Abstracts: Volume 34, Issue 4 Automorphic Forms and Representation Theory, Du- bravka Ban and Joe Hundley, Southern Illinois University, Deadlines and Shuichiro Takeda, University of Missouri, Columbia. For organizers: Expired Commutative Algebra, Lianna Sega, University of Mis- For abstracts: Expired souri, Kansas City, and Hema Srinivasan, University of Missouri, Columbia.
    [Show full text]
  • Crosscap Numbers and the Jones Polynomial
    CROSSCAP NUMBERS AND THE JONES POLYNOMIAL EFSTRATIA KALFAGIANNI AND CHRISTINE RUEY SHAN LEE Abstract. We give sharp two-sided linear bounds of the crosscap number (non-orientable genus) of alternating links in terms of their Jones polynomial. Our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings. We also discuss generalizations of our results for classes of non-alternating links. 1. Introduction The goal of this paper is to give two-sided linear bounds of the crosscap number (i.e. the non-orientable genus) of an alternating link in terms of coefficients of the Jones polynomial of the link. We show that both of these bounds are sharp and often they give the exact value of the crosscap number. As an application we calculate the crosscap number of several infinite families of alternating links. We also check that our bounds give the crosscap numbers of 283 alternating knots with up to twelve crossings that were previously unknown. Finally, we generalize our results to classes of non-alternating links. To state our results, for a link K ⊂ S3, let n n−1 ′ s+1 ′ s JK (t)= αKt + βK t + . + βK t + αKt denote the Jones polynomial of K, so that n and s denote the highest and lowest power in t. Set ′ TK := |βK| + βK , ′ where βK and βK denote the second and penultimate coefficients of JK(t), respectively. Also let sK = n − s, denote the degree span of JK (t).
    [Show full text]
  • Program of the Sessions, New Orleans, LA
    Program of the Sessions New Orleans, Louisiana, January 5–8, 2007 2:00PM D’Alembert, Clairaut and Lagrange: Euler and the Wednesday, January 3 (6) French mathematical community. Robert E. Bradley, Adelphi University AMS Short Course on Aspects of Statistical Learning, I 3:15PM Break. 3:30PM Enter, stage center: The early drama of hyperbolic 8:00 AM –4:45PM (7) functions in the age of Euler. Organizers: Cynthia Rudin, Courant Institute, New Janet Barnett, Colorado State University-Pueblo York University Miroslav Dud´ik, Princeton University 8:00AM Registration. 9:00AM Opening remarks by Cynthia Rudin and Miroslav Thursday, January 4 Dud´ik. 9:15AM Machine Learning Algorithms for Classification. MAA Board of Governors (1) Robert E. Schapire, Princeton University 10:30AM Break. 8:00 AM –5:00PM 11:00AM Occam’s Razor and Generalization Bounds. (2) Cynthia Rudin*, Center for Neural Science and AMS Short Course on Aspects of Statistical Learning, Courant Institute, New York University, and II Miroslav Dud´ik*, Princeton University 2:00PM Exact Learning of Boolean Functions and Finite 9:00 AM –1:00PM (3) Automata with Queries. Lisa Hellerstein, Polytechnic University Organizers: Cynthia Rudin, Courant Institute, New York University 3:15PM Break. Miroslav Dud´ik, Princeton University 3:45PM Panel Discussion. 9:00AM Online Learning. (8) Adam Tauman Kalai, Weizmann Institute of MAA Short Course on Leonhard Euler: Looking Back Science and Toyota Technological Institute after 300 Years, I 10:15AM Break. 10:45AM Spectral Methods for Visualization and Analysis of 8:00 AM –4:45PM (9) High Dimensional Data. Organizers: Ed Sandifer, Western Connecticut Lawrence Saul, University of California San Diego State University NOON Question and answer session.
    [Show full text]
  • Education 2015 Michigan State University, East Lansing, MI, U.S.A
    CURRICULUM VITAE Christine Ruey Shan Lee Department of mathematics, Email: [email protected] The University of Texas at Austin, Website: www.ma.utexas.edu/users/clee 2515 Speedway, RLM 8.100, Austin, TX 78712 Education 2015 Michigan State University, East Lansing, MI, U.S.A. Ph.D. in Mathematics, May 2015 Advisor: Efstratia Kalfagianni Dissertation title: Jones-type link invariants and applications to 3-manifold topology. 2008 The University of Manchester, Manchester, UK M.Sc. with distinction in Pure Mathematics, September 2008 2007 California Institute of Technology, Pasadena, CA, U.S.A. B.S. in Mathematics, June 2007 Academic Employment 2015- NSF postdoctoral fellow and instructor, University of Texas at present Austin, Austin, TX. 2017 Fall Visiting scholar, the Max Planck Institute for Mathematics, Bonn, Germany. 2009-15 Teaching and research assistant, Michigan State University, East Lansing, MI. Research Interests: Knot theory; quantum topology; 3-manifolds Publications 7. Jones slopes and coarse volume of near-alternating links, available at arXiv:1708.04900. 6. Normal and Jones surfaces of knots, with Efstratia Kalfagianni, available 1 at arXiv:1702.06466, submitted. 5. A trivial tail homology for non A-adequate links, accepted for publication in Algebraic and Geometric topology, available at arXiv:1611.00686. 4. Slopes for pretzel knots, with Roland van der Veen, New York Journal of Mathematics, 22, p. 1339{1364, 2016. 3. Stability properties of the colored Jones polynomial, available at arXiv:1409.4457, submitted. 2. Crosscap numbers and the Jones polynomial, with Efstratia Kalfagianni, Advances in Mathematics, 286, p.308-337, 2016. 1. On the degree of the colored Jones polynomial, with Efstratia Kalfagianni, Acta Mathematica Vietnamica (Proceedings of Quantum Topology and Hyperbolic Geometry in Nha Trang, May 2013), 39(4), 2014.
    [Show full text]
  • Cosets of Monodromies and Quantum Representations
    COSETS OF MONODROMIES AND QUANTUM REPRESENTATIONS RENAUD DETCHERRY AND EFSTRATIA KALFAGIANNI Abstract. We use geometric methods to show that given any 3-manifold M, and g a sufficiently large integer, the mapping class group Mod(Σg;1) contains a coset of an g abelian subgroup of rank b 2 c; consisting of pseudo-Anosov monodromies of open-book decompositions in M: We prove a similar result for rank two free cosets of Mod(Σg;1): These results have applications to a conjecture of Andersen, Masbaum and Ueno about quantum representations of surface mapping class groups. For surfaces with boundary, and large enough genus, we construct cosets of abelian and free subgroups of their mapping class groups consisting of elements that satisfy the conjecture. The mapping tori of these elements are fibered 3-manifolds that satisfy a weak form of the Turaev-Viro invariants volume conjecture. 1. Introduction It has been known since Alexander [1] that any closed, oriented 3-manifold admits an open book decomposition, and Myers [32] proved that there is one with connected binding. In recent years open book decompositions received attention as they are closely related to contact geometry through the work of Giroux [20]. For a compact oriented surface Σ := Σg;n with genus g and n boundary components, let Mod(Σ) denote the mapping class group of Σ. For f 2 Mod(Σ) and a subgroup H of Mod(Σ), we will say that fH is a rank k abelian coset if H is abelian of rank k. Similarly, we will say that fH is a free coset if H is free and non-abelian.
    [Show full text]
  • Symmetric Links and Conway Sums: Volume and Jones Polynomial
    Math. Res. Lett. 16 (2009), no. 2, 233–253 c International Press 2009 SYMMETRIC LINKS AND CONWAY SUMS: VOLUME AND JONES POLYNOMIAL David Futer, Efstratia Kalfagianni, and Jessica S. Purcell Abstract. We obtain bounds on hyperbolic volume for periodic links and Conway sums of alternating tangles. For links that are Conway sums we also bound the hyperbolic volume in terms of the coefficients of the Jones polynomial. 1. Introduction Given a combinatorial diagram of a knot in the 3–sphere, there is an associated 3–manifold, the knot complement, which decomposes into geometric pieces by work of Thurston [26]. A central goal of modern knot theory is to relate this geometric structure to simple topological properties of the knot and to combinatorial knot in- variants. To date, there are only a handful of results along these lines. Lackenby found bounds on the volume of alternating links based on the number of twist regions in the link diagram [16]. We extended these results to all links with at least seven crossings per twist region in [12], and in [11] we obtain similar results for links that are closed 3–braids. Our method is to apply a result bounding the change of volume under Dehn filling based on the length of the shortest filling slope. In all these cases the relation between twist number and volume was also important in establishing a coarse volume conjecture: a linear correlation between the coefficients of the classical Jones polynomial and the volume of hyperbolic links. In the present paper, we build upon the methods of [12] as well as very recent work of Gabai, Meyerhoff, and Milley [14]; Agol, Storm, and Thurston [8]; and Agol [6].
    [Show full text]
  • Faculty and Members 2019–2020
    Contents Introduction 2 School of Historical Studies 4 School of Mathematics 22 School of Natural Sciences 48 School of Social Science 66 Program in Interdisciplinary Studies 77 Director’s Visitors 80 Artist-in-Residence Program 83 84 Trustees and Officers of the Board and of the Corporation 86 Administration 88 Past Directors and Faculty 89 Index Information contained herein is current as of September 12, 2019 1 G41029_FacultyMembersTXT.indd 1 9/16/19 4:43 PM Introduction The Institute for Advanced Study is an international center for theoretical research and intellectual inquiry that provides an exceptional environment for the acceleration of ideas and knowledge. It creates time and space for solitary work as well as dialogue among some 250 researchers selected and mentored each year from more than 100 institutions around the world and at various stages in their careers by a permanent Faculty, each of whom are preeminent leaders in their fields. From postdocs with new perspectives and tools, to estab- lished experts who create and advance fields of inquiry, the Institute’s focused yet freely inquisitive atmosphere enables advancement in unforeseeable ways, leading to societal innovation and new understanding. Research spans four Schools—Historical Studies, Mathematics, Natural Sciences, Social Science—and is focused on long-term and fundamental outcomes with no concern for immediate application but rather revolutionary and sustained impact. IAS is a scholar’s paradise—a campus of unparalleled energy and curiosity, free of external pressures and academic restraints, where excep- tional minds have boundless opportunity to explore what is not yet known. Thirty-three Nobel Laureates, forty-two of the sixty Fields Medalists, and eighteen of the twenty Abel Prize Laureates, as well as many winners of the Wolf and MacArthur prizes, have been affiliated with the Institute.
    [Show full text]
  • View This Volume's Front and Back Matter
    CONTEMPORARY MATHEMATICS 541 Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory Workshop June 3–13, 2009 Conference June 15–19, 2009 Columbia University, New York, NY Abhijit Champanerkar, Oliver Dasbach, Efstratia Kalfagianni, Ilya Kofman, Walter Neumann, Neal Stoltzfus Editors American Mathematical Society http://dx.doi.org/10.1090/conm/541 Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory CONTEMPORARY MATHEMATICS 541 Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory Workshop June 3–13, 2009 Conference June 15–19, 2009 Columbia University, New York, NY Abhijit Champanerkar, Oliver Dasbach, Efstratia Kalfagianni, Ilya Kofman, Walter Neumann, Neal Stoltzfus Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeTurck, managing editor George Andrews Abel Klein Martin J. Strauss 2000 Mathematics Subject Classification. Primary 57Mxx, 32Qxx, 60Gxx, 16Txx, 17Bxx, 81Rxx, 81Txx, 11Sxx, 14Txx. Library of Congress Cataloging-in-Publication Data Workshop on Interactions between Hyperbolic Geometry, Quantum Topology, and Number Theory (2009 : Columbia University) Interactions between hyperbolic geometry, quantum topology, and number theory : Workshop on Interactions between Hyperbolic Geometry, Quantum Topology, and Number Theory, June 3– 13, 2009, Columbia University, New York, New York / Abhijit Champanerkar ...[et al.], editors. p. cm. Includes bibliographical references. ISBN 978-0-8218-4960-6 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Manifolds (Mathematics)—Congresses. 3. Quantum field theory—Congresses. 4. Number theory—Congresses. I. Champanerkar, Abhijit, 1975– II. Title. QA612.14.W67 2009 514.22—dc22 2010049884 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given.
    [Show full text]
  • ON 3-BRAIDS and L-SPACE KNOTS. an L-Space Knot
    ON 3-BRAIDS AND L-SPACE KNOTS. CHRISTINE RUEY SHAN LEE AND FARAMARZ VAFAEE An L-space knot generalizes the notion of knots which admit lens space surgeries. A rational homology 3-sphere Y is an L-space if jH1(Y ; Z)j = rank HFd(Y ), where HFd denotes the `hat' version of Heegaard Floer homology, and the name stems from the fact that lens spaces are L-spaces. Besides lens spaces, examples of L-spaces include all connected sums of manifolds with elliptic geometry [OS05]. A knot, K ⊂ S3, is an L-space knot if K or its mirror image admits a positive L-space surgery. One of the most prominent problems in relating low-dimensional topology and Heegaard Floer homology is to give a topological characterization to L-spaces and L-space knots. In this direction, Ozsv´athand Szab´o'sresult states that L-spaces admit no co-orientable taut foliations [OS04, Theorem 1.4]. It is also known that an L-space knot K ⊂ S3 must be prime [Krc13] and fibered [Ni07], and that K supports the tight contact structure on S3 [Hed10, Proposition 2.1]. In addition, the Alexander polynomial 4K (t) of an L-space knot K satisfies the following: • The absolute value of a nonzero coefficient of 4K (t) is 1. The set of nonzero coefficients alternates in sign [OS05, Corollary 1.3]. • If g is the maximum degree of 4K (t) in t, then the coefficients of the term tg−1 is nonzero and therefore ±1 [HW14]. The purpose of this manuscript is to study which 3-braids, that close to form a knot, admit L-space surgeries.
    [Show full text]
  • Turaev-Viro Invariants, Colored Jones Polynomials and Volume
    Turaev-Viro invariants, colored Jones polynomials and volume Renaud Detcherry, Efstratia Kalfagianni * and Tian Yang † Abstract We obtain a formula for the Turaev-Viro invariants of a link complement in terms of values of the colored Jones polynomials of the link. As an application, we give the first examples of 3-manifolds where the “large r” asymptotics of the Turaev-Viro invariants determine the hyperbolic volume. We verify the volume conjecture of Chen and the third named author [7] for the figure-eight knot and the Borromean rings. Our calculations also exhibit new phenomena of asymptotic behavior of values of the colored Jones polynomials that seem to be predicted neither by the Kashaev-Murakami- Murakami volume conjecture and its generalizations nor by Zagier’s quantum modularity conjecture. We conjecture that the asymptotics of the Turaev-Viro invariants of any link complement determine the simplicial volume of the link, and verify this conjecture for all knots with zero simplicial volume. Finally, we observe that our simplicial volume conjecture is compatible with connected summations and split unions of links. 1 Introduction In [39], Turaev and Viro defined a family of 3-manifold invariants as state sums on triangulations of manifolds. The family is indexed by an integer r; and for each r the invariant depends on a choice of a 2r-th root of unity. In the last couple of decades these invariants have been refined and generalized in many directions and shown to be closely related to the Witten-Reshetikhin-Turaev invariants. (See [2, 19, 38, 24] and references therein.) Despite these efforts, the relationship between the Turaev-Viro invariants and the geometric structures on 3-manifolds arising from Thurston’s geometrization picture is not understood.
    [Show full text]
  • Quantum Representations and Monodromies of Fibered Links
    QUANTUM REPRESENTATIONS AND MONODROMIES OF FIBERED LINKS RENAUD DETCHERRY AND EFSTRATIA KALFAGIANNI Abstract. Andersen, Masbaum and Ueno conjectured that certain quantum representa- tions of surface mapping class groups should send pseudo-Anosov mapping classes to ele- ments of infinite order (for large enough level r). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants TVr of hyperbolic 3-manifolds. We show that if the r-growth of jTVr(M)j for a hyperbolic 3-manifold M that fibers over the circle is exponential, then the monodromy of the fibration of M satisfies the AMU con- jecture. Building on earlier work [11] we give broad constructions of (oriented) hyperbolic fibered links, of arbitrarily high genus, whose SO(3)-Turaev-Viro invariants have exponen- tial r-growth. As a result, for any g > n > 2, we obtain infinite families of non-conjugate pseudo-Anosov mapping classes, acting on surfaces of genus g and n boundary components, that satisfy the AMU conjecture. We also discuss integrality properties of the traces of quantum representations and we answer a question of Chen and Yang about Turaev-Viro invariants of torus links. 1. Introduction Given a compact oriented surface Σ; possibly with boundary, the mapping class group Mod(Σ) is the group of isotopy classes of orientation preserving homeomorphisms of Σ that fix the boundary. The Witten-Reshetikhin-Turaev Topological Quantum Field Theories [24, 32] provide families of finite dimensional projective representations of mapping class groups. For each semi-simple Lie algebra, there is an associated theory and an infinite family of such representations.
    [Show full text]