Pairs of boundary slopes with small differences

TL;DR

This paper proves that for any positive real number, there exists a knot in the 3-sphere with a pair of boundary slopes differing by no more than that number, highlighting the density of boundary slope differences.

Contribution

It establishes the existence of knots with arbitrarily small differences between boundary slopes, a new result in knot theory.

Findings

Existence of knots with boundary slope differences arbitrarily close to zero

Boundary slopes can be densely packed in the 3-sphere

Advances understanding of boundary slope distributions in knot complements

Abstract

We show that, for any positive real number, there exists a knot in the 3-sphere admitting a pair of boundary slopes whose difference is at most the given number.