Postgraduate Thesis uoadl:2919730 78 Read counter

Κατεύθυνση Θεωρητικά Μαθηματικά

Library of the School of Science

Library of the School of Science

2020-07-16

2020

Manouras Manousos

Αριστείδης Κοντογιώργης, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ

FINITE TYPE INVARIANTS FOR KNOTOIDS

English

FINITE TYPE INVARIANTS FOR KNOTOIDS

In this master’s thesis the main goal is to create a theory of the finite type invariants for knotoids and to prove several central results of this theory, and especially for type-1 invariants the classification and their construction. Firstly, we introduce the diagrammatic theory of knotoids in an orientable connected surface and the basic results of the theory of knotoid invariants. Then, we introduce the finite type invariants for knots and their chord diagrams, which transform the theory of these invariants from a singular theoretic object to a combinatorial one, as well as the proof of the Vassiliev-Kontsevich theorem using the Khniznik-Zamolodchikov connections. In the central part of the thesis we create the theory of finite type invariants for knotoids and prove new results in it, such as the theorem of classification of flat singular knotoids with one singularity up to singular equivalence, by introducing the diagrammatic theory of flat singular knotoids and their linear chord diagrams. Finally, we give the construction of the universal type-1 invariant and give non-trivial examples of such invariants.

Science

knotoids, finite type invariants, knot theory, low dimensional topology

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