We prove that every possible k-cycle can be embedded into P G(n, q), for all n ≥ 3 and q a power of a prime.

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We study Agol cycles of pseudo-Anosov 3-braids using Farey sequences. We give a sufficient condition for two 3-braids to have equivalent or mirror equivalent Agol cycles and give infinitely many examples. We also study behavior of Agol cycles of pseudo-Anosov 3-braids admitting flype moves. Contents 1. Introduction 3.1. 4-tuple of triple-weight train track 3.2. Statement of main results 4. Triple-weight train track and Farey sequence 4.1. Farey sequence 4.2. The map T 5. Proof of the main theorems 6. Application to non-degenerate flypes 6.1. Birman-Menasco's classification of 3-braids 6.2. Dilatation is invariant under flypes 6.3. Mirror Agol-positive pairs 6.4. Non-conjugate braids with equivalent Agol cycles 6.5. Agol-Positive pairs 6.6. Length of Agol cycle References

Abstract. We extend the concepts of trivializing and knotting numbers for knots to spatial graphs and 2-bouquet graphs, in particular. Furthermore, we calculate the trivializing and knotting numbers for projections and pseudodiagrams of 2-bouquet spatial graphs based on the number of precrossings and the placement of the precrossings in the pseudodiagram of the spatial graph.