FU Berlin Digitale Dissertation
Alexander Stoimenow : On enumeration of chord diagrams and asymptotics of Vassiliev invariants Abzählen von Sehnendiagrammen und Asymptotik von Vassiliev-Invarianten
|Abstract| |Table of Contents| |More Information|

Abstract The subject of the present thesis are combinatorics of chord diagrams and asymptotics of Vassiliev invariants. In sections 2 and 3 we will derive some (purely) enumerative results on special kinds of chord diagrams. Although not directly related to Vassiliev invariants, these results provide a glimpse of the combinatorial complexity of chord diagrams -- already for easily to define properties the enumeration is rather hard and requires additional ideas. Parts of this work can be found in several papers of mine In section 4 we will use combinatorial techniques to relate enumeration of special chord diagrams to a context of Vassiliev invariants and will prove the asymptotical upper bound $D!/1.1^D$ for the number of Vassiliev invariants in the degree $D$. 5 we will use the techniques of section 4 and the result of Chmutov and Duzhin to deduce a lower bound for the number of all Vassiliev invariants and discuss the relation between the asymptotics of prime and all Vassiliev invariants. Parallely, we give a summary on what we know about the asymptotics of Vassiliev invariants. Finally, in section 6 we use the rather different approach of braiding sequences to prove exponential upper bounds for the number of Vassiliev invariants on knots with bounded braid index and arborescent knots.

Table of Contents Download the whole PhDthesis as a zip-tar file or as zip-File For download in PDF format click the chapter title 1. Vassiliev Invariants for knots 1 1.1 The classification problem of knots 4 1.2 The filtration of the knot space 4 1.3 The Algebra A 6 1.4 Weight systems 7 1.5 VASSILIEV invariants for braids and string links 8 1.6 Constructing a universal VASSILIEV invariant 9 1.7 Braiding sequences 9 2. The results of this thesis 3. On the number of chord diagrams 3.1 Notations 11 3.2 Linearized chord diagrams 11 3.3 Cyclic CD´s and GLCD´s 12 3.4 Counting all chord diagrams 14 3.5 Symmetric chord diagrams 14 3.6 Degenerate CD´s and LCD´s 15 3.7 Chord diagrams with chords of length 1 18 3.8 Chord diagrams with isolated chords only 19 3.9 Some computations 20 3.10 Asymptotics 20 4. Connected and tree-connected chord diagrams 4.1 Connected CD´s and LCD´s 22 4.2 Tree--connected CD´s and LCD´s 24 4.3 Some computations 27 5. An upper bound for Vassiliev invariants 5.1 Factoring out 4T relations 27 5.2 Regular linearized chord diagrams 29 5.3 Connected regular LCD´s 32 5.4 Numerical and asymptotical results 34 5.5 A further improvement 38 5.6 The segment length inequality 43 6. The dimension of a commutative graded algebra and asymptotics of VI 6.1 The dominating partition 43 6.2 A lower bound for the number of all Vassiliev invariantss 45 6.3 The exponential barrier 46 7. The braid index and the growth of Vassiliev invariants 7.1 Braiding sequences 47 7.2 Arborescent knots 48 7.3 Bounds for braid representations 51 7.4 The growth of the number of knots and Vassiliev invariantss 54 References B Abstract 59 A Zusammenfassung (German abstract) 59

More Information:
Online available:	http://www.diss.fu-berlin.de/1999/21/indexe.html
Language of PhDThesis:	english
Keywords:	Vassiliev invariants, chord diagrams, upper bound, braids, arborescent knots, partitions
DNB-Sachgruppe:	27 Mathematik
Classification MSC:	57M25, 57M15
Date of disputation:	06-May-1998
PhDThesis from:	Fachbereich Mathematik u. Informatik, Freie Universität Berlin
First Referee:	Prof. Dr. Elmar Vogt
Second Referee:	Prof. Dr. S. Chmutov
Contact (Author):	stoimeno@hp832.informatik.hu-berlin.de
Contact (Advisor):	vogt@math.fu-berlin.de
Date created:	14-Apr-1999
Date available:	14-Apr-1999

 

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