Difference between revisions of "Knot table"
From Encyclopedia of Mathematics
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| − | The list of diagrams of all simple knots admitting a projection on the plane with 9 or fewer double points. The notation for the knots in this table is standard; the first number indicates the number of double points and the second (placed as a suffix) the ordinal number of the knot. E.g., the knot | + | {{TEX|done}} |
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| + | The list of diagrams of all simple knots admitting a projection on the plane with 9 or fewer double points. The notation for the knots in this table is standard; the first number indicates the number of double points and the second (placed as a suffix) the ordinal number of the knot. E.g., the knot $7_5$ is the fifth knot in the table with 7 intersections. Alongside each knot in coded form is given its Alexander polynomial $\Delta(t)=a_{2n}t^{2n}+\dots+a_nt^n+\dots+a_0$ (cf. [[Alexander invariants|Alexander invariants]]). Since the Alexander polynomial of every knot has even degree and is reciprocal (i.e. $a_i=a_{2n-i}$), it suffices to give the set of last coefficients $a_n,\dots,a_0$; they are indicated in the table. E.g., next to the knot $8_9$ is written $7-5+3-1$. This means that the Alexander polynomial equals $\Delta(t)=-t^6+3t^5-5t^4+7t^3-5t^2+3t-1$. Non-alternating knots are marked by an asterisk (cf. [[Alternating knots and links|Alternating knots and links]]). The table is taken from {{Cite|a1}} with minor modifications. | ||
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/k055590a.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/k055590a.gif" /> | ||
Figure: k055590a | Figure: k055590a | ||
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====Comments==== | ====Comments==== | ||
| − | A table of knots with up to 10 crossings can be found in | + | A table of knots with up to 10 crossings can be found in {{Cite|a2}}. |
====References==== | ====References==== | ||
| − | + | * {{Ref|a1}} G. Burde, "Knoten" , ''Jahrbuch Ueberblicke Mathematik'' , B.I. Wissenschaftsverlag Mannheim (1978) pp. 131–147 | |
| + | * {{Ref|a2}} D. Rolfsen, "Knots and links" , Publish or Perish (1976)</TD></TR></table> | ||
Latest revision as of 08:08, 17 March 2023
2020 Mathematics Subject Classification: Primary: 57K [MSN][ZBL]
The list of diagrams of all simple knots admitting a projection on the plane with 9 or fewer double points. The notation for the knots in this table is standard; the first number indicates the number of double points and the second (placed as a suffix) the ordinal number of the knot. E.g., the knot $7_5$ is the fifth knot in the table with 7 intersections. Alongside each knot in coded form is given its Alexander polynomial $\Delta(t)=a_{2n}t^{2n}+\dots+a_nt^n+\dots+a_0$ (cf. Alexander invariants). Since the Alexander polynomial of every knot has even degree and is reciprocal (i.e. $a_i=a_{2n-i}$), it suffices to give the set of last coefficients $a_n,\dots,a_0$; they are indicated in the table. E.g., next to the knot $8_9$ is written $7-5+3-1$. This means that the Alexander polynomial equals $\Delta(t)=-t^6+3t^5-5t^4+7t^3-5t^2+3t-1$. Non-alternating knots are marked by an asterisk (cf. Alternating knots and links). The table is taken from [a1] with minor modifications.
Figure: k055590a
Comments
A table of knots with up to 10 crossings can be found in [a2].
References
- [a1] G. Burde, "Knoten" , Jahrbuch Ueberblicke Mathematik , B.I. Wissenschaftsverlag Mannheim (1978) pp. 131–147
- [a2] D. Rolfsen, "Knots and links" , Publish or Perish (1976)
How to Cite This Entry:
Knot table. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Knot_table&oldid=15052
Knot table. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Knot_table&oldid=15052
This article was adapted from an original article by M.Sh. Farber (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article