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Neuwirth knot

From Encyclopedia of Mathematics
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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)
How to Cite This Entry:
Neuwirth knot. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neuwirth_knot&oldid=55805
This article was adapted from an original article by M.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article