Generalizations of Quandles to Multi-Linkoids

by Runa Pflume
Master thesis
Date of Examination:2023-11-17
Date of issue:2024-04-10
Advisor:Dr. Neslihan Gügümcü
Referee:Prof. Dr. Thomas Schick
Referee:Dr. Neslihan Gügümcü
Persistent Address: http://dx.doi.org/10.53846/goediss-10442

 

 

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Abstract

English

In this thesis, we define the fundamental quandle of knotoids and linkoids and prove that it is invariant under the under forbidden-move and hence encodes only the information of the underclosure of the knotoid. We then introduce $n$-pointed quandles, which generalize quandles by specifying $n$ elements as ordered basepoints. This leads to the notion of fundamental pointed quandles of linkoids, which enhances the fundamental quandle. Using 2-pointed quandle allows us to distinguish 1-linkoids with equivalent under-closures and leads to a couple of 1-linkoid invariants. In particular we study implications on the 2-cocyle invariant. We then define $n$-pointed biquandles in a similar way to use biquandle colorings to distinguish 1-linkoids. We also generalize the notion of homogeneity of quandles to $n$-homogeneity of quandles. We classify all $\infty$-homogeneous, finite quandles.
Keywords: knots; linkoids; knotoids; knotoid quandle; linkoid quandle
 
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