Dokument

Summary:
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.

Bibliographie / References

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25. Remark 1.4.2. Since (x imm(ΣHi2 ) ≤ npii22 = npii11−−22++132 ≤ nni1i1−−22++21+23 = 4(ni1−2+12) 4 ni1−2+14 If Hj1−2 ≺1 Hj1−1 ≺1 Hj1, then nj1 ≤ nj1−2 + 6. Now consider Figure 3.3.20, where vj1 ∈ Vxy such that vj1 ∈ Hj1 \ Hj1−1 and Hj1−2 ≺2 Hj1−1 ≺1 Hj1.
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