Triple linking numbers and triple point numbers of certain $T^2$-links

TL;DR

This paper introduces a method to compute the triple linking number of certain $T^2$-links using linking numbers of associated braids, providing bounds and exact values for triple point numbers in specific cases.

Contribution

It presents a formula for the triple linking number of $T^2$-links derived from commutative pure braids, linking it to braid linking numbers and determining some triple point numbers.

Findings

Triple linking number expressed via braid linking numbers.

Lower bounds for triple point numbers established.

Exact triple point numbers determined in specific cases.

Abstract

The triple linking number of an oriented surface link was defined as an analogical notion of the linking number of a classical link. We consider a certain $m$-component $T^2$-link ($m \geq 3$) determined from two commutative pure $m$-braids $a$ and $b$. We present the triple linking number of such a $T^2$-link, by using the linking numbers of the closures of $a$ and $b$. This gives a lower bound of the triple point number. In some cases, we can determine the triple point numbers, each of which is a multiple of four.