View source for Neuwirth knot

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A polynomial knot $(S^3,k^1)$ (cf. [[Knot theory|Knot theory]]) whose group has a finitely-generated commutator subgroup. The complement $S^3\setminus k^1$ of a Neuwirth knot is a [[Fibre space|fibre space]] over a circle and the fibre $F$ is a connected surface whose genus is that of the knot. The [[Commutator subgroup|commutator subgroup]] $G'$ of the group $G=\pi_1(S^3\setminus k^1)$ of a Neuwirth knot is a free group of rank $2g$, where $g$ is the genus of the knot. The coefficient of the leading term of the Alexander polynomial of a Neuwirth knot (cf. [[Alexander invariants|Alexander invariants]]) is 1 and the degree of this polynomial is $2g$. All torus knots (cf. [[Torus knot|Torus knot]]) are Neuwirth knots. So is every alternating knot whose Alexander polynomial has leading coefficient $\pm1$.

These knots were introduced by L. Neuwirth (see [[#References|[1]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.P. Neuwirth,   "Knot groups" , Princeton Univ. Press  (1965)</TD></TR></table>