Barvanja spletov in Goeritzeva matrika

Bogataj, Lucija (author), Strle, Sašo (mentor)

V magistrskem delu obravnavamo vozle in splete v prostoru ter njihove diagrame. Na diagrame spletov ob izbrani Abelovi grupi $A$ vpeljemo Foxova barvanja, Dehnova barvanja, šahovsko senčenje in Goeritzevo matriko. Pokažemo povezave med temi objekti. Grupe Dehnovih barvanj različnih diagramov istega spleta v prostoru so med seboj izomorfne. Prav tako grupe Foxovih barvanj. Pokažemo, kakšna je struktura grupe Foxovih barvanj in struktura grupe Dehnovih barvanj. Obe sta izomorfni direktni vsoti jedra Goeritzeve matrike in ustreznega števila ponovitev grupe $A$, le da je grupa Dehnovih barvanj za en $A$ razsežnejša od grupe Foxovih barvanj.

• Language: Slovenian
• Keywords: vozel, splet, Goeritzeva matrika, Foxovo barvanje, Dehnovo barvanje
• Work type: Master's thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• Publication date in RUL: 20.07.2024

Secondary language

• Language: English
• Title: Link colorings and the Goeritz matrix

Abstract

In the master's thesis, we discuss knots and links in space and their diagrams. Fox colorings, Dehn colorings, checkerboard shading, and Goeritz matrix are applied to link diagrams considering the selected Abelian group $A$. We show the connections between these objects. Groups of Dehn colorings of different diagrams of the same link in space are mutually isomorphic. Also groups of Fox colorings. We show the structure of the group of Fox colorings and the structure of the group of Dehn colorings. Both are isomorphic to the direct sum of the kernel of the Goeritz matrix and an appropriate number of copies of the group $A$, except that the group of Dehn colorings contains one more copy of $A$ than the group of Fox colorings.

• Keywords: knot, link, Goeritz matrix, Fox coloring, Dehn coloring