Twist knot

In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite family of knots, and are considered the simplest type of knots after the torus knots.

A twist knot is obtained by linking together the two ends of a twisted loop. Any number of half-twists may be introduced into the loop before linking, resulting in an infinite family of possibilities. The following figures show the first few twist knots:

One half-twist
(trefoil knot)

Two half-twists
(figure-eight knot)

Three half-twists
(5₂ knot)

Four half-twists
(stevedore knot)

Five half-twists
(7₂ knot)

Six half-twists
(8₁ knot)

All twist knots have unknotting number one, since the knot can be untied by unlinking the two ends. Every twist knot is also a 2-bridge knot. Of the twist knots, only the unknot and the stevedore knot are slice knots. A twist knot with ${\displaystyle n}$ half-twists has crossing number ${\displaystyle n+2}$. All twist knots are invertible, but the only amphichiral twist knots are the unknot and the figure-eight knot.

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