L9a33

L9a32

L9a34

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a33 at Knotilus!
	L9a33 is $9_{24}^{2}$ in the Rolfsen table of links.

Symmetric form		Alternate symmetric version, with three lines touching at center		Alternate symmetric version, with three lines touching at circumference
Form made from 45-degree lines and circular arcs.		Depiction obtained by knotilus.		With an hypotrochoid [1].
Mexican book.

Link Presentations

[edit Notes on L9a33's Link Presentations]

Planar diagram presentation	X8192 X12,3,13,4 X18,10,7,9 X10,14,11,13 X16,5,17,6 X14,18,15,17 X2738 X4,11,5,12 X6,15,1,16
Gauss code	{1, -7, 2, -8, 5, -9}, {7, -1, 3, -4, 8, -2, 4, -6, 9, -5, 6, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {u^{2}v^{2}-3u^{2}v+3u^{2}-3uv^{2}+7uv-3u+3v^{2}-3v+1}{uv}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {7}{q^{7/2}}}+q^{5/2}-{\frac {9}{q^{5/2}}}-4q^{3/2}+{\frac {10}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+6{\sqrt {q}}-{\frac {8}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-3a^{5}z-a^{5}z^{-1}+3a^{3}z^{3}+3a^{3}z-az^{5}-2az^{3}+z^{3}a^{-1}-3az$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-3a^{5}z^{7}-7a^{3}z^{7}-4az^{7}-2a^{6}z^{6}-7a^{4}z^{6}-11a^{2}z^{6}-6z^{6}-a^{7}z^{5}+5a^{5}z^{5}+9a^{3}z^{5}-az^{5}-4z^{5}a^{-1}+3a^{6}z^{4}+18a^{4}z^{4}+24a^{2}z^{4}-z^{4}a^{-2}+8z^{4}+3a^{7}z^{3}-3a^{5}z^{3}-2a^{3}z^{3}+8az^{3}+4z^{3}a^{-1}-11a^{4}z^{2}-14a^{2}z^{2}-3z^{2}-3a^{7}z+a^{5}z+2a^{3}z-2az-a^{6}+a^{7}z^{-1}+a^{5}z^{-1}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 3 χ 6                   1 -1 4                 3   3 2               3 1   -2 0             5 3     2 -2           6 4       -2 -4         3 4         -1 -6       4 6           2 -8     2 3             -1 -10     4               4 -12 1 2                 -1 -14 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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