On the Indeterminacy of the Triple Linking Number

Abstract

In the 1950’s Milnor defined a new family of tools of link theory generalizing the classical linking number. When the classical linking numbers vanish, the first of these new invariants μ123(L) gives new interesting information. In the case that the classical linking numbers are non-zero, Milnor’s invariants are not well defined. In recent work, Davis-Nagel-Orson-Powell defined new invariants called the total triple linking number and the total Milnor quotient that improved on the triple linking number. In this project we compute the total Milnor quotient and show that it is non-trivial for every link of at least six components. Thus, the total triple linking can be used to distinguish links even when none of the triple linking numbers are well defined.