Coexistence of coiled surfaces and spanning surfaces for knots and links

TL;DR

This paper generalizes a classical construction of torus knots and links using closed fake surfaces, proving the resulting surfaces are essential and relating boundary slope distance to triple points, with implications for the Neuwirth conjecture.

Contribution

It introduces a new method using closed fake surfaces to construct knots and surfaces, proving their essentiality and linking boundary slope distance to triple points.

Findings

The constructed surfaces are essential.

Knots satisfy the Neuwirth conjecture.

Boundary slope distance equals the number of triple points.

Abstract

It is a well-known procedure for constructing a torus knot or link that first we prepare an unknotted torus and meridian disks in the complementary solid tori of it, and second smooth the intersections of the boundary of meridian disks uniformly. Then we obtain a torus knot or link on the unknotted torus and its Seifert surface made of meridian disks. In the present paper, we generalize this procedure by a closed fake surface and show that the resultant two surfaces obtained by smoothing triple points uniformly are essential. We also show that a knot obtained by this procedure satisfies the Neuwirth conjecture, and the distance of two boundary slopes for the knot is equal to the number of triple points of the closed fake surface.