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

  • Surgery presentations for knots coloured by metabelian groups
    See arXiv:1101.0532.
    A G-coloured knot is a knot together with a representation of its knot group onto G. Two G-coloured knots are said to be rho-equivalent if they are related by surgery around unit framed unknots in the kernels of their colourings. The induced local move is a G-coloured analogue of the crossing change. For certain families of metabelian groups G, we classify G-coloured knots up to rho-equivalence. Our method involves passing to a problem about G-coloured analogues of Seifert matrices.

    Work in Progress

  • A Kontsevich Invariant for Coloured Knots
    Using our surgery presentation of a D_{2n}-coloured knot, we construct a non-commutative version of the rational Kontsevich invariant for D_{2n}-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 D_{2n}-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    .

    Courses Taught

  • MAT137- Calculus! University of Toronto, Autumn-Spring 2010-2011.
  • MAT1900- Dehn Surgery. University of Toronto, Summer 2011.
  • MAT137- Calculus! University of Toronto, Autumn-Spring 2011-2012.
  • MAT332- Introduction to Graph Theory. University of Toronto, Autumn 2011.
  • MAT224- Linear Algebra II. University of Toronto, Mississauga, Spring 2012.

    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.
  • First steps in coloured knot theory
    Topology Seminar, University of Victoria, Victoria, Canada, January 28 2011.
  • First steps in coloured knot theory
    Special session on "Topological, Geometric, and Quantum Invariants of 3-Manifolds", Spring Eastern Sectional Meeting, Worcester, Massachusetts, USA, April 10 2011.

    Miscellaneous

  • My favourite books
  • Useful LaTeX resources
  • Proofs of the Kirby theorem
  • Miscellaneous mathematical stuff
  • Some homepages
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彟楯影捰㹴ㄽ〰簠⁼慍桴昮潬牯䴨瑡⹨慲摮浯⤨ㄪ〰⠯〱ⴰ彟楯影捰⥴
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੻††晩⠠椡噳
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‿摡慃⹴浤穯㨠✠敭扭牥⤧਻ †椠⁦琨楨⹳祬潣彳敳牡档煟敵祲
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੻††††††桴獩氮捹獯慟孤汳瑯⁝‽摡杍⹲敧却潬⡴汳瑯㬩 †††素 †素ਊ †愠䵤牧爮湥敤䡲慥敤⡲㬩 †愠䵤牧爮湥敤䙲潯整⡲㬩紊⠨畦据楴湯⤨笠 †瘠牡眠㴠〠‬⁨‽ⰰ洠湩浩浵桔敲桳汯⁤‽〳㬰 †椠⁦琨灯㴠‽敳晬
੻††††敲畴湲琠畲㭥 †素ਊ††晩⠠祴数景眨湩潤⹷湩敮坲摩桴
㴽✠畮扭牥‧
੻††††⁷‽楷摮睯椮湮牥楗瑤㭨 †††栠㴠眠湩潤⹷湩敮䡲楥桧㭴 †素 †攠獬⁥晩⠠潤畣敭瑮搮捯浵湥䕴敬敭瑮☠…搨捯浵湥⹴潤畣敭瑮汅浥湥⹴汣敩瑮楗瑤⁨籼搠捯浵湥⹴潤畣敭瑮汅浥湥⹴汣敩瑮效杩瑨⤩笠 †††眠㴠搠捯浵湥⹴潤畣敭瑮汅浥湥⹴汣敩瑮楗瑤㭨 †††栠㴠搠捯浵湥⹴潤畣敭瑮汅浥湥⹴汣敩瑮效杩瑨਻††੽††汥敳椠⁦搨捯浵湥⹴潢祤☠…搨捯浵湥⹴潢祤挮楬湥坴摩桴簠⁼潤畣敭瑮戮摯⹹汣敩瑮效杩瑨⤩笠 †††眠㴠搠捯浵湥⹴潢祤挮楬湥坴摩桴਻††††⁨‽潤畣敭瑮戮摯⹹汣敩瑮效杩瑨਻††੽ †爠瑥牵⠨⁷‾業楮畭呭牨獥潨摬
☦⠠⁨‾業楮畭呭牨獥潨摬⤩਻⡽⤩⤩਻ਊ眊湩潤⹷湯潬摡㴠映湵瑣潩⡮
੻††慶⁲⁦‽潤畣敭瑮朮瑥汅浥湥䉴䥹⡤氢捹獯潆瑯牥摁⤢਻††慶⁲⁢‽潤畣敭瑮朮瑥汅浥湥獴祂慔乧浡⡥戢摯≹嬩崰਻††⹢灡数摮桃汩⡤⥦਻††⹦瑳汹⹥楤灳慬⁹‽戢潬正㬢 †搠捯浵湥⹴敧䕴敬敭瑮祂摉✨祬潣䙳潯整䅲楤牆浡❥⸩牳⁣‽⼧摡⽭摡是潯整䅲⹤晩慲敭栮浴❬਻ †⼠ 汓摩牥䤠橮捥楴湯 †⠠畦据楴湯⤨笠 †††瘠牡攠㴠搠捯浵湥⹴牣慥整汅浥湥⡴椧牦浡❥㬩 †††攠献祴敬戮牯敤⁲‽〧㬧 †††攠献祴敬洮牡楧‽㬰 †††攠献祴敬搮獩汰祡㴠✠汢捯❫਻††††⹥瑳汹⹥獣䙳潬瑡㴠✠楲桧❴਻††††⹥瑳汹⹥敨杩瑨㴠✠㔲瀴❸਻††††⹥瑳汹⹥癯牥汦睯㴠✠楨摤湥㬧 †††攠献祴敬瀮摡楤杮㴠〠਻††††⹥瑳汹⹥楷瑤⁨‽㌧〰硰㬧 †素⠩㬩ਊ †⼠ 潂瑴浯䄠⁤湉敪瑣潩੮††
畦据楴湯⤨笠 †††瘠牡戠㴠搠捯浵湥⹴敧䕴敬敭瑮䉳呹条慎敭∨潢祤⤢せ㭝ਊ††††慶⁲楩⁦‽潤畣敭瑮挮敲瑡䕥敬敭瑮✨晩慲敭⤧਻††††楩⹦瑳汹⹥潢摲牥㴠✠✰਻††††楩⹦瑳汹⹥慭杲湩㴠〠਻††††楩⹦瑳汹⹥楤灳慬⁹‽戧潬正㬧 †††椠晩献祴敬挮獳汆慯⁴‽爧杩瑨㬧 †††椠晩献祴敬栮楥桧⁴‽㈧㐵硰㬧 †††椠晩献祴敬漮敶晲潬⁷‽栧摩敤❮਻††††楩⹦瑳汹⹥慰摤湩⁧‽㬰 †††椠晩献祴敬眮摩桴㴠✠〳瀰❸਻††††楩⹦牳⁣‽⼧摡⽭摡椯橮捥䅴⹤晩慲敭栮浴❬਻†††† †††瘠牡挠楤⁶‽潤畣敭瑮挮敲瑡䕥敬敭瑮✨楤❶㬩 †††挠楤⹶瑳汹⁥‽眢摩桴㌺〰硰活牡楧㩮〱硰愠瑵㭯㬢 †††挠楤⹶灡数摮桃汩⡤椠晩⤠਻††††晩
⁢਩††††੻††††††⹢湩敳瑲敂潦敲挨楤ⱶ戠氮獡䍴楨摬㬩 †††素 †素⠩㬩ਊ੽ਊ⼼捳楲瑰ਾ㰊瑳汹㹥⌊潢祤⸠摡敃瑮牥汃獡⁳੻†慭杲湩〺愠瑵㭯 搠獩汰祡戺潬正℠浩潰瑲湡㭴 漠敶晲潬㩷楨摤湥਻†楷瑤㩨〱┰਻੽戣摯⁹愮䍤湥整䍲慬獳⌠摡损湯慴湩牥笠 搠獩汰祡戺潬正℠浩潰瑲湡㭴 映潬瑡氺晥㭴 眠摩桴㜺㠲硰਻੽浀摥慩⠠業⵮楷瑤㩨㜠㠶硰
੻†ℼⴭ䘠牯㌠〰硰漠⁲敬獳愠獤传䱎⁙ⴭਾ†戣摯⁹愮䍤湥整䍲慬獳⌠摡损湯慴湩牥笠 †眠摩桴›慣捬ㄨ〰‥‭㜳瀲⥸਻†੽੽浀摥慩⠠業⵮楷瑤㩨ㄠㄱ瀰⥸笠 㰠ⴡ‭潆⁲㈷瀸⁸牯氠獥⁳摡⁳ⴭਾ†戣摯⁹愮䍤湥整䍲慬獳⌠摡损湯慴湩牥笠 †眠摩桴›慣捬ㄨ〰‥‭㜳瀲⥸਻†੽੽㰊猯祴敬ਾ㰊楤⁶瑳汹㵥戢捡杫潲湵㩤愣敢昶㬶戠牯敤⵲潢瑴浯ㄺ硰猠汯摩⌠〵愷㜸※潰楳楴湯爺汥瑡癩㭥稠椭摮硥㤺㤹㤹㤹㸢ਊ††搼癩挠慬獳∽摡敃瑮牥汃獡≳ਾ††††愼栠敲㵦栢瑴獰⼺眯睷愮杮汥楦敲氮捹獯挮浯∯琠瑩敬∽湁敧晬物⹥潣㩭戠極摬礠畯⁲牦敥眠扥楳整琠摯祡∡猠祴敬∽楤灳慬㩹汢捯㭫映潬瑡氺晥㭴眠摩桴ㄺ㘸硰※潢摲牥〺㸢 ††††㰠浩⁧牳㵣⼢摡⽭摡愯杮汥楦敲昭敲䅥⹤灪≧愠瑬∽楓整栠獯整⁤祢䄠杮汥楦敲挮浯›畂汩⁤潹牵映敲⁥敷獢瑩⁥潴慤ⅹ•瑳汹㵥搢獩汰祡戺潬正※潢摲牥〺•㸯 †††㰠愯ਾ††††搼癩椠㵤愢彤潣瑮楡敮≲ਾ††††††猼牣灩⁴祴数∽整瑸樯癡獡牣灩≴搾捯浵湥⹴牷瑩⡥祬潣彳摡❛敬摡牥潢牡❤⥝㰻猯牣灩㹴 †††㰠搯癩ਾ††⼼楤㹶㰊搯癩ਾ㰊ⴡ‭⼯⼯⼯⼯⼯⼯⼯⼯⼯⼯⼯⼯⼯⼯⼯⼯⼯⼯ ⴭਾ猼牣灩⁴祴数∽整瑸樯癡獡牣灩≴搾捯浵湥⹴牷瑩⡥祬潣彳摡❛汳摩牥崧㬩⼼捳楲瑰ਾਊ搼癩椠㵤氢捹獯潆瑯牥摁•瑳汹㵥戢捡杫潲湵㩤愣敢昶㬶戠牯敤⵲潴㩰瀱⁸潳楬⁤㔣㜰㡡㬷挠敬牡戺瑯㭨搠獩汰祡渺湯㭥瀠獯瑩潩㩮敲慬楴敶※⵺湩敤㩸㤹㤹㤹∹ਾ搼癩挠慬獳∽摡敃瑮牥汃獡≳猠祴敬∽楤灳慬㩹汢捯Ⅻ浩潰瑲湡㭴漠敶晲潬㩷楨摤湥※楷瑤㩨㌹瀶㭸㸢ऊ搼癩椠㵤愢汦湩獫潨摬牥•瑳汹㵥昢潬瑡氺晥㭴眠摩桴ㄺ㘸硰∻ਾ††††愼栠敲㵦栢瑴獰⼺眯睷愮杮汥楦敲氮捹獯挮浯∯琠瑩敬∽湁敧晬物⹥潣㩭戠極摬礠畯⁲牦敥眠扥楳整琠摯祡∡猠祴敬∽楤灳慬㩹汢捯㭫戠牯敤㩲∰ਾ††††††椼杭猠捲∽愯浤愯⽤湡敧晬物ⵥ牦敥摁⸲灪≧愠瑬∽楓整栠獯整⁤祢䄠杮汥楦敲挮浯›畂汩⁤潹牵映敲⁥敷獢瑩⁥潴慤ⅹ•瑳汹㵥搢獩汰祡戺潬正※潢摲牥〺•㸯 †††㰠愯ਾ††⼼楤㹶 †㰠晩慲敭椠㵤氢捹獯潆瑯牥摁䙩慲敭•瑳汹㵥戢牯敤㩲㬰搠獩汰祡戺潬正※汦慯㩴敬瑦※敨杩瑨㤺瀶㭸漠敶晲潬㩷楨摤湥※慰摤湩㩧㬰眠摩桴㜺〵硰㸢⼼晩慲敭ਾ⼼楤㹶㰊搯癩ਾਊℼⴭ‭乕䕄䑒䝏䕍䥄⁁䑅䕇江捹獯挮浯䨠癡卡牣灩⁴䑁佃䕄匠䅔呒ⴭ㸭㰊捳楲瑰搠瑡ⵡ晣獡湹㵣昢污敳•慬杮慵敧∽慪慶捳楲瑰•獡湹⁣牳㵣⼢甯浤敳癲⹥敮⽴摵⽭浩⹧敦捴㽨楳㵤㜱㔷㬴楴㵤㬱瑤㘽∻㰾猯牣灩㹴㰊ⴡⴭ唠䑎剅佄䵇䑅䅉䔠䝄彅祬潣⹳潣慊慶捓楲瑰䄠䍄䑏⁅久ⵄⴭਾ