= Trefoil knot
{title2=$3_1$}
{wiki}
The trefoil is the <torus knot> $T_{2,3}$. It is the simplest nontrivial <knot> by <crossing number of a knot>: $c(T_{2,3})=3$. A standard three-crossing <alternating knot diagram> is reduced, so the <Tait crossing-number theorem> establishes minimality. The four-crossing <knot diagram> shown below also represents the <trefoil knot>, illustrating that a particular projection need not achieve that minimum. Its <Seifert genus> is one.
Here is a small algebraic certificate that a <trefoil knot> cannot be deformed into an <unknot>. One <Seifert matrix> for a <trefoil knot> is
$$
A=\begin{pmatrix}1&0\\-1&1\end{pmatrix},\qquad tA-A^T=\begin{pmatrix}t-1&1\\-t&t-1\end{pmatrix}.
$$
Taking the <determinant> gives the <Alexander polynomial of a knot>
$$
\det(tA-A^T)=(t-1)^2+t=t^2-t+1\doteq t-1+t^{-1},
$$
where $\doteq$ allows multiplication by a <Laurent unit> $\pm t^n$ in the <Laurent polynomial ring>. No such multiplication turns this three-term <Alexander polynomial of a knot> into the <unknot>'s value $1$. Since the <Alexander polynomial of a knot> is a <knot invariant>, this computation rules out every possible <ambient isotopy> between the two <knots>; it does not merely rule out the deformations we happened to try.
The <knot group> supplies another certificate. The presentation $\langle a,b\mid a^2=b^3\rangle$ maps onto the <symmetric group> $S_3$ by $a\mapsto(12)$ and $b\mapsto(123)$, since both sides of the relation map to the identity. These images generate a <non-abelian group>, so the <knot group> cannot be the <unknot>'s <infinite cyclic group>. However, a <trefoil knot> and its <mirror of a link> have isomorphic <knot groups>: the abstract <knot group> alone does not distinguish all <knots>.
Back to article page
