Difference between revisions of "Neuwirth knot"
From Encyclopedia of Mathematics
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| − | A polynomial knot | + | {{TEX|done}} |
| + | 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]]]). | These knots were introduced by L. Neuwirth (see [[#References|[1]]]). | ||
Revision as of 20:34, 11 April 2014
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=16263
Neuwirth knot. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neuwirth_knot&oldid=16263
This article was adapted from an original article by M.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article