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Abstract
In this thesis we give two applications of Alexander ideals to knotted surfaces in $S^4$. First we prove that the Alexander ideal induces a homomorphism from the 0-concordance monoid $\mathscr{C}_0$ of oriented surface knots in $S^4$ to the ideal class monoid of $\Zt$. Consequently, any surface knot with nonprincipal Alexander ideal is not 0-slice and in fact, not invertible in $\mathscr{C}_0$. This proves that the submonoid of 2-knots is not a group and reproves the existence of infinitely many linearly independent 0-concordance classes.
The second application is to regular homotopies of 2-knots in $S^4$, and is joint work with Michael Klug, Benjamin Ruppik, and Hannah Schwartz. Analogous to classical unknotting number, we define the Casson-Whitney number of a 2-knot as the minimal number of Whitney moves during any regular homotopy to the unknot, and prove that if $K_1$ and $K_2$ each have nontrivial determinant, then the Casson-Whitney number of $K_1\# K_2$ is at least 2. A corollary is that the Casson-Whitney number is not equal to the stabilization number, the minimal number of 1-handle stabilizations needed to produce an unknotted surface. We also prove a strong version of nonadditivity for both the Casson-Whitney number and the stabilization number.
The second application is to regular homotopies of 2-knots in $S^4$, and is joint work with Michael Klug, Benjamin Ruppik, and Hannah Schwartz. Analogous to classical unknotting number, we define the Casson-Whitney number of a 2-knot as the minimal number of Whitney moves during any regular homotopy to the unknot, and prove that if $K_1$ and $K_2$ each have nontrivial determinant, then the Casson-Whitney number of $K_1\# K_2$ is at least 2. A corollary is that the Casson-Whitney number is not equal to the stabilization number, the minimal number of 1-handle stabilizations needed to produce an unknotted surface. We also prove a strong version of nonadditivity for both the Casson-Whitney number and the stabilization number.