Unknot (source code)
= Unknot
{title2=$U$}
{wiki}
The unknot $U$ is the <knot> type represented by the planar <circle>
$$
U(t)=(\cos t,\sin t,0),\qquad t\in\mathbb R/(2\pi\mathbb Z).
$$
Equivalently, an <unknot> is a <knot> that bounds a smoothly embedded <disk> in the <three-sphere>. The <disk> must itself be embedded; merely filling a loop with a map that is allowed to intersect itself does not establish that the <knot> is an <unknot>. Flattening a neighborhood of an embedded spanning <disk> explains why this criterion recovers the planar <circle>.
A complicated drawing can still represent an <unknot>. Starting with a crossing-free <knot diagram> of $U$, insert a small kink by a type I <Reidemeister move>. The resulting <knot diagram> has one crossing, but removing the kink exhibits the same <unknot>. Repeating in separate small regions gives <knot diagrams> of the <unknot> with arbitrarily many crossings, although the <crossing number of a knot> is $c(U)=0$. A drawing's apparent complexity is therefore not a certificate of knotting.
The normalized <Alexander polynomial of a knot> and <Jones polynomial> of the <unknot> are both $1$, and its <knot group> is the <infinite cyclic group> $\mathbb Z$, because the <knot exterior> is a <solid torus>. These are useful comparison values. In particular, a nonconstant <Alexander polynomial of a knot> certifies that a <knot> is not an <unknot>, but the converse fails: nontrivial untwisted <Whitehead doubles> also have <Alexander polynomial of a knot> $1$.
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