Quandles and Hurwitz Orbits
In this thesis we study quandles and Hurwitz orbits. A quandle is a self-distributive algebraic structure whose binary operation is like the conjugation in a group. The algebraic structure of quandles can be studied as sequences of permutations. The cycle structure of the permutations of an indec...
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| Format: | Dissertation |
| Sprache: | Englisch |
| Veröffentlicht: |
Philipps-Universität Marburg
2016
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| Zusammenfassung: | In this thesis we study quandles and Hurwitz orbits. A quandle is a self-distributive
algebraic structure whose binary operation is like the conjugation in a group. The algebraic
structure of quandles can be studied as sequences of permutations. The cycle
structure of the permutations of an indecomposable quandle is well-behaved because
the permutations of an indecomposable quandle are mutually conjugate and hence have
the same cycle structure. We study the cycle structure of quandles with the main focus
on a conjecture in [18] saying that any permutation of an indecomposable quandle has
cycles whose cycle lengths divide the largest among them.
Hurwitz orbits are the orbits of a braid group action on the powers of a quandle. The
Hurwitz orbits for the action of the braid group on three strands are used in [21] and
[22] for the classiVcation of certain Hopf algebras. This classiVcation is based on a combinatorial
invariant called a plague on the Hurwitz orbits. The immunity on a Hurwitz
orbit is the quotient of the size of the minimal plague and the size of the Hurwitz orbit.
An estimation on the immunity of the Hurwitz orbits is provided in [22] by using
Schreier graphs of the Hurwitz orbit quotients and the weights on the Hurwitz orbits,
where the weight on an Hurwitz orbit is deVned by the cycle structure of that Hurwitz
obit. In this study only few Schreier graphs of the Hurwitz orbit quotients with small
cycles are considered. We introduce a new method to calculate plagues on the Hurwitz
orbits for inVnitely many Schreier graphs of the Hurwitz orbit quotients with all cycles.
Our method is based on the posets of robust subgraphs of pointed Schreier graphs of the
Hurwitz orbit quotients. By using this method we estimate the immunity on the Hurwitz
orbits through a case-by-case analysis of inVnitely many pointed Schreier graphs
of the Hurwitz orbit quotients. |
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| Umfang: | 86 Seiten |
| DOI: | 10.17192/z2016.0652 |