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: Image:One-Twist Trefoil.png|One half-twist ([[trefoil knot]], 31) Image:Blue Figure-Eight Knot.png|Two half-twists ([[Figure-eight knot (mathematics)|figure-eight knot]], 41) Image:Blue Three-Twist Knot.png|Three half-twists ([[Three-twist knot|52 knot]]) Image:Blue Stevedore Knot.png|Four half-twists ([[Stevedore knot (mathematics)|stevedore knot]], 61) Image:Blue 7_2 Knot.png|Five half-twists (72 knot) Image:Blue 8_1 Knot.png|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 half-twists has crossing number . All twist knots are invertible, but the only amphichiral twist knots are the unknot and the figure-eight knot. The invariants of a twist knot depend on the number of half-twists.