In this thesis, we present two results in different areas of Knot Theory. We show that the tunnel of a tunnel number one, fibered link can be isotoped into the fiber as an arc that is clean under the monodromy. This constitutes progress in an effort to show that every tunnel number one, fibered link can be constructed from the unknot by a sequence of Hopf plumbings along tunnels in fibers. We also show that every closed, orientable three-manifold admits a knot whose complement has a Heegaard splitting of arbitrarily high Hempel distance. Relatedly, we show that handlebody disk sets are coarsely distinct in the complex of curves.
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