Narrow Results

We consider the question of when is the closed manifold obtained by elementary surgery on an n-knot Seifert fibered over a 2-orbifold. The possible bases are strongly constrained by the fact that knot groups have weight 1. We present a new family of 2-knots with solvable groups, overlooked in earlier work. The knots in this new family are neither invertible nor amphicheiral, and the weight orbits for the knot groups are parametrized by ℤ. There are no known examples in higher dimensions.

A twisted torus knot is a knot obtained from a torus knot by twisting adjacent strands by full twists. The twisted torus knots lie in $F$, the genus 2 Heegaard surface for $S^3$. Primitive/primitive and primitive/Seifert knots lie in $F$ in a particular way. Dean gives sufficient conditions for the parameters of the twisted torus knots to ensure they are primitive/primitive or primitive/Seifert. Using Dean’s conditions, Doleshal shows that there are infinitely many twisted torus knots that are fibered and that there are twisted torus knots with distinct primitive/Seifert representatives with the same slope in $F$. In this paper, we extend Doleshal’s results to show there is a four parameter family of positive twisted torus knots. Additionally, we provide new examples of twisted torus knots with distinct representatives with the same surface slope in $F$.