The Tait conjectures concern reduced alternating knot diagrams: they minimize the crossing number of a knot, have an invariant diagram writhe, and are related by flypes. These formerly conjectural assertions are now theorems; the first is the Tait crossing-number theorem.
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The Tait conjectures, proposed by the Scottish mathematician Peter Guthrie Tait in the late 19th century, relate to the field of knot theory, a branch of topology. Tait conjectured that there are specific relationships between the number of crossings in a knot diagram and its properties, particularly concerning its link or knot type.