Date of this Version
V. Longo. On 2-knots and connected sums with projective planes. Geom.
Dedicata, 207:23–27, 2020.
A classical knot is a smooth embedding of the circle into the 3-sphere. We can also consider embeddings of arbitrary surfaces (possibly nonorientable) into a 4-manifold, called knotted surfaces. In this thesis, we give an introduction to some of the basics of the studies of classical knots and knotted surfaces, then present some results about nonorientable surfaces bounded by classical knots and embeddings of nonorientable knotted surfaces. First, we generalize a result of Satoh about connected sums of projective planes and twist spun knots. Specifically, we will show that for any odd natural n, the connected sum of the n-twist spun sphere of a knot K and an unknotted projective plane in the 4-sphere becomes equivalent to the same unknotted projective plane after a single trivial stabilization. We additionally provide a fix to a small error in Satoh's proof of the case that K is a 2-bridge knot. Additionally, we show that the band unknotting number of a classical knot is an upper bound for the unknotting number of any twist spin of the classical knot. This result is also motivated by a result of Satoh which states that for a classical knot, the unknotting number and the bridge number minus one are both upper bounds for the twist spin of the classical knot. Milnor's conjecture, first proved by Kronheimer and Mrowka in 1993, states that the 4-ball genus of a torus knot T(p,q) is equal to 1/2(p-1)(q-1). Batson's conjecture is a nonorientable version of Milnor's conjecture which states that the nonorientable 4-ball genus is equal to the pinch number of a torus knot, i.e. the number of a specific type of (nonorientable) band surgeries needed to obtain the unknot. The conjecture was recently proved to be false by Lobb. We will show that Lobb's counterexample fits into an infinite family of counterexamples.
Advisor: Alex Zupan and Mark Brittenham