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Author
Title Classical Lie Algebra Weight Systems of Arrow Diagrams
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Publication Date
Date Available
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
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) 7 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

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