Full Record

Author | Leung, Louis |

Title | Classical Lie Algebra Weight Systems of Arrow Diagrams |

URL | http://hdl.handle.net/1807/26366 |

Publication Date | 2010 |

Date Available | 2011-02-23 00:00:00 |

Degree Level | doctoral |

University/Publisher | University of Toronto |

Abstract | The notion of finite type invariants of virtual knots, introduced by Goussarov, Polyak and Viro, leads to the study of the space of diagrams with directed chords mod 6T (also known as the space of arrow diagrams), and weight systems on it. It is well known that given a Manin triple together with a representation we can construct a weight system. In the first part of this thesis we develop combinatorial formulae for weight systems coming from standard Manin triple structures on the classical Lie algebras and these structures' defining representations. These formulae reduce the problem of finding weight systems in the defining representations to certain counting problems. We then use these formulae to verify that such weight systems, composed with the averaging map, give us the weight systems found by Bar-Natan on (undirected) chord diagrams mod 4T. In the second half of the thesis we present results from computations done jointly with Bar-Natan. We compute, up to degree 4, the dimensions of the spaces of arrow diagrams whose skeleton is a line, and the ranks of all classical Lie algebra weight systems in all representations. The computations give us a measure of how well classical Lie algebras capture the spaces of arrow diagrams up to degree 4, and our results suggest that in degree 4 there are already weight systems which do not come from the standard Manin triple structures on classical Lie algebras. PhD |

Subjects/Keywords | knot theory; virtual knots; weight systems; 0405 |

Contributors | Bar-Natan, Dror; Mathematics |

Language | en |

Country of Publication | ca |

Record ID | handle:1807/26366 |

Repository | toronto-diss |

Date Retrieved | 2020-03-09 |

Date Indexed | 2020-03-09 |

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…weight systems and finite type invariants of oriented *virtual* knots modulo “braid-like”
Reidemeister moves (see [BHLR] and below), which are Reidemeister moves where the
part of the *knot* involved is locally a braid. We say an invariant…

…of *virtual* knots is of
type n if it vanishes on all *virtual* *knot* diagrams with more than n semi-*virtual* crossings.
(The smallest such n is called the degree of the invariant.) A semi-*virtual* crossing is the
difference between a real crossing…

…given a *virtual*
*knot* diagram we can always express it as a linear combination of *virtual* *knot* diagrams
with only semi-*virtual* and *virtual* crossings. Equivalently, given a Gauss diagram with
solid arrows we can always turn it into a linear combination of…

…*knot* diagrams modulo braid-like Reidemeister II and braid-like Reidemeister III.
Invariants of type n of braid-like *virtual* knots are those which vanish on diagrams with
8
more than n semi-*virtual* crossings.
By the result of the computations presented…

…1.2
Relations between finite type invariants and weight
systems
This section is a review of the notion of finite type invariants of *virtual* knots and corresponding weight systems introduced in [GPV]. Also we consider the relation between
5…

…and a *virtual* crossing. On the level of Gauss diagrams
we use solid arrows to represent real crossings and dotted arrows to represent semi-*virtual*
crossings (figure 1.8). An invariant is said to be of finite type if it is of type n for some
n…

…the following definition.
Definition 1.2.1. The space of (long) braid-like *virtual* knots is the space of (long)
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Figure 1.11. Reidemeister II in terms of Gauss diagrams for any sign σ. For
braid-like Reidemeister II both strands…

…orientations give us braid-like Reidemeister III.
Figure 1.13. A different way of drawing 6T. The 6T relation can be obtained from
the braid-like 8T relation where all arrows have the same signs by modding out by
degree-(n + 1) diagrams.
*virtual*…