Difference between revisions of "Listing knot"
From Encyclopedia of Mathematics
(Importing text file) |
(svg picture) |
||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| − | One of the simplest non-trivial knots (see Fig. and [[Knot theory|Knot theory]]). A Listing knot is denoted by the symbol | + | {{TEX|done}} |
| + | One of the simplest non-trivial knots (see Fig. and [[Knot theory|Knot theory]]). A Listing knot is denoted by the symbol $4_1$ (see [[Knot table|Knot table]]) and is sometimes called a figure 8 or fourfold knot. The group of the Listing knot (cf. [[Knot and link groups|Knot and link groups]]) has the presentation $|x,y\colon yx^{-1}yxy^{-1}=x^{-1}yxy^{-1}x|$, and the Alexander polynomial is $\Delta_1=t^2-3t+1$. It was considered by I.B. Listing [[#References|[1]]]. | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l059730a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/l059730a.gif" /> | ||
Figure: l059730a | Figure: l059730a | ||
| + | |||
| + | [[File:Listing knot.svg|center|100px|Listing knot projection]] | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table> |
| − | + | <TR><TD valign="top">[1]</TD> <TD valign="top"> I.B. Listing, "Vorstudien zur Topologie", Göttingen (1847)</TD></TR> | |
| − | + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> R.H. Crowell, R.H. Fox, "Introduction to knot theory", Ginn (1963)</TD></TR> | |
| − | + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> L.H. Kauffman, "On knots", Princeton Univ. Press (1987)</TD></TR> | |
| − | + | </table> | |
| − | + | {{OldImage}} | |
| − | |||
| − | |||
Latest revision as of 18:02, 22 April 2023
One of the simplest non-trivial knots (see Fig. and Knot theory). A Listing knot is denoted by the symbol $4_1$ (see Knot table) and is sometimes called a figure 8 or fourfold knot. The group of the Listing knot (cf. Knot and link groups) has the presentation $|x,y\colon yx^{-1}yxy^{-1}=x^{-1}yxy^{-1}x|$, and the Alexander polynomial is $\Delta_1=t^2-3t+1$. It was considered by I.B. Listing [1].
Figure: l059730a
References
| [1] | I.B. Listing, "Vorstudien zur Topologie", Göttingen (1847) |
| [a1] | R.H. Crowell, R.H. Fox, "Introduction to knot theory", Ginn (1963) |
| [a2] | L.H. Kauffman, "On knots", Princeton Univ. Press (1987) |
🛠️ This page contains images that should be replaced by better images in the SVG file format. 🛠️
How to Cite This Entry:
Listing knot. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Listing_knot&oldid=18188
Listing knot. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Listing_knot&oldid=18188
This article was adapted from an original article by M.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article