Narrow Results

Let K be a classical knot in ℝ³. We can deform the diagram of K to that of a trivial knot by changing the overcrossings and undercrossings at some double points of the diagram of K. We consider the same problem for higher dimensinal knots. Let n≥2 and π:ℝⁿ⁺²=ℝⁿ⁺¹×ℝ→ℝⁿ⁺¹ denote the natural projection map. A pseudo-ribbon n-knot is an n-knot f:Sⁿ→ℝⁿ⁺² such that the self-intersection set of π◦f:Sⁿ→ℝⁿ⁺¹ consists of only double points and is homeomorphic to a disjoint union of (n-1)-spheres. We prove that for n≠3,4, the projection (π◦f)(Sⁿ)⊂ℝⁿ⁺¹ of any pseudo-ribbon n-knot f is the projection of a trivial n-knot.

There are exactly four mutually non-isotopic unknotting tunnels τ_i, i = 1,2,3,4 for the pretzel knot P(-2,3,7). Moreover, there are at most 3 non-stabilized genus 3 Heegaard splittings.

Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the minimum number of generators of the first homology of the double branched cover of $S^3$ over the knot is strictly less than the multi-region index. Our proof of this lower bound uses Goeritz matrices.

This paper defines a new operation through extending the idea of the 0-dimensional crossing change and Shimizu’s 2-dimensional region crossing change [A. Shimizu, Region crossing change is an unknotting operation, J. Math. Soc. Jpn. 66(3) (2014) 693–708, doi:10.2969/jmsj/06630693] to a 1-dimensional version called the arc crossing change. We will also prove that the arc crossing change is an unknotting operation with the help of Gauss diagrams.