A New Proof That Alternating Links Are Non-Trivial

Abstract

We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. Unlike most other proofs of this result, this proof uses only classic results in topology and combinatorial group theory.

Graph Small Cancellation Theory Applied to Alternating Link Groups

“…We give a correspondance between the two viewpoints: if Γ satisfies the non metric conditions C(p) − T (q) O ("O" for "Ollivier"), then each megatile diagram of a minimal diagram is a [p, q]-map, in the terminology of [4]. Then we obtain an analogous result to the main technical result of classical small cancellation theory in the non-metric case (see [4] theorem 6.3 p 262 and theorem 7.6 p 265) giving solutions to the word and conjugacy problems, from the graph viewpoint: see theorem 5.…”

“…It is also clear that if a prime alternating presentation P (Γ, R) of a knot contains at least one crossing, then the knot is not trivial, as it is easy to show that the shortest relation has length 4, and all the generators of P (Γ, R) are meridians or their inverses (see [5]). If the group were Z, then there would be a relation of length 2.…”