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:
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.[1] Of the twist knots, only the unknot and the stevedore knot are slice knots. A twist knot with
n
n+2
The invariants of a twist knot depend on the number
n
\Delta(t)=\begin{cases}
n+1 | |
2 |
t-n+
n+1 | |
2 |
t-1&ifnisodd\\ -
n | |
2 |
t+(n+1)-
n | |
2 |
t-1&ifniseven,\\ \end{cases}
and the Conway polynomial is
\nabla(z)=\begin{cases}
n+1 | |
2 |
z2+1&ifnisodd\\ 1-
n | |
2 |
z2&ifniseven.\\ \end{cases}
When
n
V(q)=
1+q-2+q-n-q-n-3 | |
q+1 |
,
and when
n
V(q)=
q3+q-q3-n+q-n | |
q+1 |
.