On the Structure and Scarcity of Alternating Knots

Abstract

Given a class of objects, a pattern theorem is a powerful result describing their structure. We show that alternating knots exhibit a pattern theorem, and use this result to prove a long-standing conjecture that alternating knots grow rare. This is currently the best possible analogue of a pair of theorems on alternating links of Sundberg and Thistlethwaite in 1998 and Thistlethwaite in 1998, given the current obstructions to an exact enumeration of knot diagrams. We also discuss implications of this pattern theorem for subknots and slipknots in minimal alternating knot diagrams and types, partially answering a conjecture of Millett and Jablan.