Narrow Results

We study knots in 𝕊³ obtained by the intersection of a minimal surface in ℝ⁴ with a small 3-sphere centered at a branch point. We construct new examples of minimal knots. In particular we show the existence of non-fibered minimal knots. We show that simple minimal knots are either reversible or fully amphicheiral; this yields an obstruction for a given knot to be a simple minimal knot. Properties and invariants of these knots such as the algebraic crossing number of a braid representative and the Alexander polynomial are studied.

We define new families of (so called multi-#) local moves on knot projections (which contain the #-local move and the ordinary crossing change) and study some of their properties together with related knot invariants. We show that they define unknotting operations and hence resulting unknotting numbers. We use 4-manifold theory as a tool and more specifically P. Gilmer's thesis ([3]).