# Neuwirth knot

A polynomial knot $(S^3,k^1)$ (cf. 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 over a circle and the fibre $F$ is a connected surface whose genus is that of the knot. The 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) is 1 and the degree of this polynomial is $2g$. All torus knots (cf. 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 [1]).

#### References

[1] | L.P. Neuwirth, "Knot groups" , Princeton Univ. Press (1965) Zbl 0184.48903 |

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Neuwirth knot.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Neuwirth_knot&oldid=55805