Daniel Moskovich

Post-Doctoral Researcher
Department of Mathematics
University of Toronto
e-mail: dmoskovich@[remove-me]gmail.com

General


I am a postdoc working on quantum topology of knots and 3-manifolds, although most of what I'm actually doing is fairly classical. I'm currently working with Dror Bar-Natan at the University of Toronto. I blog on mathematics that interests me at Low Dimensional Topology.


Publications

  • Surgery presentations of coloured knots and of their covering links (Joint with A.J. Kricker). Alg. Geom. Topol. 9 (2009), 1341-1398.
    See arXiv:math.GT/0805.2307
    We consider knots equipped with a representation of their knot groups onto a dihedral group D_{2n} (where n is odd). To each such knot there corresponds a closed 3-manifold, the (irregular) dihedral branched covering space, with the branching set over the knot forming a link in it. We report a variety of results relating to the problem of passing from the initial data of a D_{2n}-coloured knot to a surgery presentation of the corresponding branched covering space and covering link. In particular, we describe effective algorithms for constructing such presentations. A by-product of these investigations is a proof of the conjecture that two D_{2n}-coloured knots are related by a sequence of surgeries along unit-framed unknots in the kernel of the representation if and only if they have the same coloured untying invariant (a Z_{n}-valued algebraic invariant of D_{2n}-coloured knots).

  • Notes on Yoshida's Coordinates on Hitchin's Prym Cover (Joint with S.K. Hansen). Acta Math. Vietnam. 33(3) (2008), 291-320.
    See HERE.
    As the first stage of his proposed geometric quantization of the SU(2) WZW model, T. Yoshida introduced coordinates on a Prym variety which covers the moduli space of semi-stable rank 2 holomorphic vector bundles with trivial determinant over a Riemann surface. We explain Yoshida's coordinates, and reprove their key properties using elementary combinatorial arguments.

  • Vanishing of 3-loop Jacobi diagrams of odd degree (Joint with T. Ohtsuki). J. Combin. Theory Ser. A. 114 (2007), 919-931.
    See arXiv:math.GT/0511602
    We prove the vanishing of the space of 3-loop Jacobi diagrams of odd degree. This implies that no three-loop finite-type invariant can distinguish between a knot and its inverse.

  • Acyclic Jacobi Diagrams Kobe J. Math. 23 (2006), 29-50.
    See arXiv:math.QA/0507351
    We propose a simple new combinatorial model to study spaces of acyclic Jacobi diagrams, in which they are identified with algebras of words modulo operations. This provides a starting point for a word-problem type combinatorial investigation of such spaces, and provides fresh insights on known results.

  • Surgery Untying of Coloured Knots Alg. Geom. Topol. 6 (2006), 673-697.
    See arXiv:math.GT/0506541
    For p=3 and for p=5 we prove that there are exactly p equivalence classes of p-coloured knots modulo 1-framed and -1-framed surgeries along unknots in the kernel of a p-colouring. These equivalence classes are represented by connect-sums of n left-hand (p,2)-torus knots with a given colouring when n=1,2,...,p. This gives a 3-colour and a 5-colour analogue of the surgery presentation of a knot.

  • Framing and the Self-Linking Integral Far East J. Math Sci 14(2) (2004), 165-183.
    See arXiv:math.GT/0211223 (This was a reading project under the supervision of Dror Bar-Natan)
    The Gauss self-linking integral of an unframed knot is not a knot invariant, but it can be turned into an invariant by adding a correction term which requires adding extra structure to the knot. We collect the different definitions/theorems/proofs concerning this correction term, most of which are well-known (at least as folklore) and put everything together in an accessible format. We then show simply and elegantly how these approaches coincide.

    Preprints


    Work in Progress

  • A Kontsevich Invariant for Coloured Knots
    Using our non-commutative surgery presentation of a knot in, we construct a non-commutative version of the rational Kontsevich invariant for p-coloured knots as a dihedrally-equivariant invariant of their irregular dihedral covering spaces. As part of the construction we prove a non-commutative analogue of the Kirby theorem for untying links of p-coloured knots. Draft version.

  • Untying Metabelian Coloured Knots
    We prove that any relative bordism between knots coloured by metabelian groups with abelianization Z can be upgraded to a surgery presentation, and provide a way of determining whether they are relative bordant. Thus, the only obstruction to obtaining a surgery presentation of such a coloured knot in the kernel of the colouring of another is the obvious one. This sets surgery-theoretic groundwork for a proposed theory of quantum topology for such coloured knots.

    Translations

  • Doubles Mélanges des Polylogarithmes Multiples aux Racines de l'Unité By Georges Racinet. Translation from French.
    A translation of Georges Racinet's landmark paper relating shuffle relations of multiple zeta functions to Drinfel'd's associator. My French is very bad so there may be mistakes, and corrections are most welcome!
    To download it click HERE    .

  • Fibrés de Rang 2 sur une Courbe, Fibré Déterminant et Fonctions Thêta By Arnaud Beauville. Translation from French.
    A translation of Arnaud Beauville's paper quoted by T. Yoshida. Again, corrections are most welcome!
    To download it click HERE    .

    Seminars Organized

  • Yoshida's Abelianization (informal seminar)- joint with K. Ueda. At RIMS, 2005.
    We worked together on understanding Yoshida's proposed abelianization of the WZW model.
  • Low dimensional topology- joint with P.A. Gastesi and J.J. Zuniga. At TIFR, 2008.
    I gave a talk on Wajnryb's MCG presentation, and 4 talks introducing knot thoery.

    Lecture Notes

    Lectures on Topology of Words - by Vladimir Turaev.
    Notes taken jointly with Eri Hatakenaka and Tadayuki Watanabe.

    Talks Given

  • Framing and the Self-Linking Integral
    KOOK Seminar, Osaka City Univesity, Algebra and Geometry of Knots and Manifolds I, August 23-26 2003.
  • A Combinatorial Calculus for $\mathcal{A}$-Spaces
    East Asian School of Knots, Links, and Related Topics, Seoul, February 16-20 2004.
  • Symmetrizing Vassiliev Invariants of Links
    International Workshop for Graduate Students about Knot Theory and Related Topics, Osaka City University, July 5-7 2004.
  • A Surgery Presentation for Irregular Branched Dihedral Covering Spaces of Knots
    III Joint Meeting Japan-Mexico in Topology and its Applications, Oaxaca, December 6-10 2004.
  • Presenting p-Coloured Knots by Links in the Kernel of the Colouring of a (p,2)-Torus Knot
    2005 International General Topology Symposium in Zhangzhou, May 25-28 2005.
  • Coloured Untying of Knots
    Osaka University, Low Dimensional Topology Seminar, Osaka University, July 17 2005.
  • A Surgery Presentation for 3-Coloured Knots and for 5-Coloured Knots
    KOOK Seminar, Algebra and Geometry of Knots and Manifolds III, Kobe, August 29-September 1 2005.
    To download the notes click HERE.
    To download the article for the proceedings click HERE.
  • A Kontsevich Integral for Fox Coloured Knots
    NZ-Japan Knot Theory conference. University of Auckland, January 4-7 2006.
    To download the notes click HERE.
  • Quantum Topology for Coloured Knots
    Geometry and Topology Seminar, University of Copenhagen, November 6 2006.
  • Vanishing of the Space of 3-Loop Jacobi Diagrams of Odd Degree
    Workshop- Geometry, Dynamics, and Complex Analysis, Schaeffersgarden, Gentofte, September 24-25 2006.
  • A Non-Commutative Analogue of the Rational Kontsevich Integral
    Topology Seminar, Aarhus University, January 30 2007.
  • Finite Type Invariants of Knots
    Departmental Colloquium, Technical University of Denmark, February 28 2007.
  • Yoshida's Abelianization Explained
    International Conference on Quantum Topology, Institute of Mathematics, VAST, Hanoi, August 6-10 2007.
  • Two Surgery Presentations for Dihedral Covering Spaces
    Friday Seminar on Knot Theory, Osaka City University, Osaka, Novermber 30 2007.
  • Surgery Presentation for Dihedral Covering Links
    The Fourth East Asian School of Knots and Related Topics, January 21-24 2008.
    To download the slides click HERE.
  • Towards surgery presentations of metabelian coloured knots and their covering links
    Friday Seminar on Knot Theory, Osaka City University, Osaka, May 30 2008.
    To download the slides click HERE.
  • Surgery Presentations for Coloured Knots and for their Covering Links
    Geometry and Topology Seminar, Indian Institute of Technology Bombay , Mumbai, India, August 20 2008.
  • Surgery Presentations for Coloured Knots and for their Covering Links
    Departmental Colloquium, Tata Institute for Fundamental Research, Mumbai, India, August 21 2008.
  • Surgery Equivalence Classes of Knots Coloured by Metabelian Groups
    The Mathematics of Knots: Theory and Application, Heidelberg, Germany, December 15-19, 2008.
    To download the slides click HERE.
  • An Alexander polynomial for coloured knots
    The 5th East Asian School of Knots and Related Topics, Gyeongju, Korea, January 11–16 2009.
  • Equivalence relations generated by surgeries which preserve metabelian information
    Friday Seminar on Knot Theory, Osaka City University, Osaka, Japan, April 24 2009.
  • Surgery presentations for metabelian-group-coloured knots
    RIMS Postdoc Seminar, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, June 6 2009.
  • Surgery presentations for knots coloured by metabelian groups
    Workshop on Topology and Geometry--- Quandles and Related Topics, Hiroshima University, Hiroshima, Japan, July 11-12 2009.
  • Surgery and bordism for coloured knots
    University of California, Berkeley, October 21 2009; University of Nevada, Reno, October 28 2009; Indiana University, November 3 2009; University of Toronto, November 6 2009; Brandeis University, November 10 2009; Columbia University, November 13 2009.
  • Symmetric surgery presentations for symmetric manifolds
    The 6th East Asian School of Knots and Related Topics, Chern Institute of Mathematics, Nankai University, Tianjin, China, January 25-28 2010.
  • Untying coloured knots
    Geometry and Topology Seminar, KAIST, Daejeon, South Korea, March 9 2010.
  • Untying coloured knots
    RIMS Postdoc Seminar, Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan, April 8 2010.
  • Untying coloured knots
    MS Seminar, Institute for the Physics and Mathematic of the Universe, Kashiwa, Japan, May 11 2010.

    Miscellaneous

  • My favourite books
  • Useful LaTeX resources
  • Proofs of the Kirby theorem
  • Some homepages