Difference between revisions of "Listing knot"
From Encyclopedia of Mathematics
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| − | 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" /> | ||
Revision as of 08:36, 12 April 2014
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) |
Comments
References
| [a1] | R.H. Crowell, R.H. Fox, "Introduction to knot theory" , Ginn (1963) |
| [a2] | L.H. Kauffman, "On knots" , Princeton Univ. Press (1987) |
How to Cite This Entry:
Listing knot. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Listing_knot&oldid=31597
Listing knot. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Listing_knot&oldid=31597
This article was adapted from an original article by M.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article