Height, trunk and representativity of knots

TL;DR

This paper explores geometrical invariants of knots, providing counterexamples, bounds, and distinctions among different knot positions, advancing understanding of knot complexity and structure.

Contribution

It introduces a counterexample to the additivity of height, establishes bounds relating representativity and trunk, and discusses differences among knot positions.

Findings

Counterexample to height additivity under connected sum

Representativity is bounded by half the trunk

Existence of knots with large representativity despite non-orientable surfaces

Abstract

In this paper, we investigate three geometrical invariants of knots, the height, the trunk and the representativity. First, we give a conterexample for the conjecture which states that the height is additive under connected sum of knots. We also define the minimal height of a knot and give a potential example which has a gap between the height and the minimal height. Next, we show that the representativity is bounded above by a half of the trunk. We also define the trunk of a tangle and show that if a knot has an essential tangle decomposition, then the representativity is bounded above by half of the trunk of either of the two tangles. Finally, we remark on the difference among Gabai's thin position, ordered thin position and minimal critical position. We also give an example of a knot which bounds an essential non-orientable spanning surface, but has arbitrarily large representativity.