Surface links with free abelian link groups

TL;DR

This paper constructs examples of torus links with free abelian link groups of rank three or four, exploring their properties and demonstrating the diversity of such links beyond previously known limitations.

Contribution

It provides explicit examples of T^2-links with free abelian link groups of higher rank and analyzes their triple point numbers and link types.

Findings

Constructed T^2-links with free abelian link groups of rank three and four.

Determined the triple point numbers for rank three T^2-links.

Showed infinitely many link types for rank three T^2-links.

Abstract

It is known that if a classical link group is a free abelian group, then its rank is at most two. It is also known that a $k$-component 2-link group ($k>1$) is not free abelian. In this paper, we give examples of $T^2$-links each of whose link groups is a free abelian group of rank three or four. Concerning the $T^2$-links of rank three, we determine the triple point numbers and we see that their link types are infinitely many.