Daniel Moskovich

Post-Doctoral Researcher
School of Mathematics
TATA Institute of Fundamental Research
e-mail: dmoskovich@[remove-me]gmail.com

General


I am a mathematian working on quantum topology of knots and 3-manifolds. I am currently a visiting fellow at the School of Mathematics of the TATA Institute of Fundamental Research in Mumbai, India.


Publications

  • 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.
  • 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.
  • 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.
  • 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.

    Preprints

  • Surgery Description of Orientation-Preserving D_{2p}-Actions on Compact Orientable 3-Manifolds.
    See arXiv:math.GT/0508141
    In analogy with Przytycki and Sokolov's ``surgery description'' of orientation-preserving periodic diffeomorphisms of closed orientable 3-manifolds, we show that there is a ``surgery description'' of orientation-preserving D_{2p}-periodic diffeomorphisms of 3-manifolds which are regular p-fold covers of the 3-sphere, for p=3, 5. This is a direct corollary of the main theorem of our "Coloured Untying of Knots" paper when plugged into the first four sections of a paper of Makoto Sakuma's, generalizing where necessary.
  • Notes on Yoshida's Coordinates on Hitchin's Prym Cover.(Joint with S.K. Hansen). Submitted.
    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 vector bundles with trivial determinant over a Riemann surface. We explain Yoshida's coordinates, and reprove their key properties using elementary combinatorial arguments.
    To download it click HERE    .
  • Surgery presentations of coloured knots and of their covering links (Joint with A.J. Kricker). Submitted.
    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).

    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.
    To download it click HERE    .
  • Surgery Presentations of Metabelian Covering Spaces

    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

    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, Januray 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.

    Miscellaneous

  • My favourite books
  • Useful LaTeX resources
  • Some homepages