singleknotted

Singleknotted is an informal term used in knot theory to describe a knot that can be transformed into the unknot by a single crossing change. More formally, such knots are said to have unknotting number one. A crossing change is the local operation of switching an overcrossing to an undercrossing (or vice versa) at a single crossing in a knot diagram. If a knot has unknotting number one, a single crossing change suffices to untie it, and no fewer changes can achieve the same result.

Examples and scope: The trefoil knot, the simplest nontrivial knot, is a standard example with unknotting number one, making it a canonical instance of singleknotted behavior. Many other low-crossing knots also have unknotting number one, though determining the exact unknotting number can be challenging for more complex knots.

Relation to other concepts: Unknotting number is a distinct measure from other knot invariants such as genus or slice genus. It is not determined by the crossing number alone. Various invariants and techniques—including determinant, signature, and polynomials—can provide obstructions to a knot having unknotting number one, but no single invariant fully characterizes the class.

Notation and usage: While singleknotted is used informally in some discussions, the standard terminology in formal literature is that a knot has unknotting number one. The term highlights the idea of “knottedness” that can be removed with just one local modification.

See also: unknotting number.