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Difference between revisions of "Neuwirth knot"

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A polynomial knot <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066470/n0664701.png" /> (cf. [[Knot theory|Knot theory]]) whose group has a finitely-generated commutator subgroup. The complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066470/n0664702.png" /> of a Neuwirth knot is a [[Fibre space|fibre space]] over a circle and the fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066470/n0664703.png" /> is a connected surface whose genus is that of the knot. The [[Commutator subgroup|commutator subgroup]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066470/n0664704.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066470/n0664705.png" /> of a Neuwirth knot is a free group of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066470/n0664706.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066470/n0664707.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066470/n0664708.png" />. All torus knots (cf. [[Torus knot|Torus knot]]) are Neuwirth knots. So is every alternating knot whose Alexander polynomial has leading coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066470/n0664709.png" />.
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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]]]).
 
These knots were introduced by L. Neuwirth (see [[#References|[1]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.P. Neuwirth,  "Knot groups" , Princeton Univ. Press  (1965)</TD></TR></table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  L.P. Neuwirth,  "Knot groups" , Princeton Univ. Press  (1965) {{ZBL|0184.48903}}</TD></TR>
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</table>

Latest revision as of 11:37, 1 June 2024

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
How to Cite This Entry:
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