Tait's conjecture on minimal crossing diagrams of links

Let $L$ be a link. A diagram of $L$ is reduced alternating if it is alternating and has no nugatory crossings. The crossing number of $L$ is the minimum number of crossings among all diagrams of $L$.

Tait's conjecture. A reduced alternating diagram of $L$ has minimal crossing number, and if $L$ is prime then no non-alternating diagram has minimal crossing number.

This conjecture initiated the study of minimal crossing diagrams of links and was proved independently by Thistlethwaite, Kauffman, and Murasugi in 1987.