# The Asymmetry Number of Finite Tournaments, and Some Related Results

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A cycle C of a graph Î“ embedded in a 3-manifold M is said to be trivial in Î“ if it bounds a disk with interior disjoint from Î“. Let e be an edge of Î“ with ends on C. We shall study the relation between triviality of cycles in Î“ and that of Î“ âˆ’ e and Î“/e. Let C1...