Concept

Free group

The group of all reduced words in a set of letters and their inverses, with no relations beyond a letter cancelling against its own inverse. It is the fundamental group of a wedge of circles, and every group generated by that many elements is a quotient of it.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A wedge of 2 circles. Several circles all passing through one common point, each labelled with a generator, so that a loop is a word in those letters.

The subgroup that is freer than the group

A free group on two letters contains a subgroup of index three that is free on four. Nothing about a group makes that plausible; everything about a graph makes it obvious, and the argument is to stop looking at the group and start looking at the space whose loops it is.

topology · covering spaces

the Borromean rings, with every crossing signed. A diagram of the Borromean rings with the under-strand broken at each crossing and each crossing between two components marked with its sign, which add to twice the linking number.

Linked, and no two of them are

Three rings that cannot be pulled apart, in which every pair comes apart the moment the third is removed. Every pairwise linking number is zero, so the number cannot see it — and what does see it is a word in two letters that refuses to cancel.

topology · linking number

Named alongside it

The objects these essays reach for when they reach for this one.

Fundamental group Borromean rings Brunnian link Commutator Covering space Generator Graph Index Linking number Loop Milnor invariant Rank

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