L9a33

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

[edit L9a33 Further Notes and Views]

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 X_12,3,13,4 X_18,10,7,9 X_10,14,11,13 X_16,5,17,6 X_14,18,15,17 X₂₇₃₈ X_4,11,5,12 X_6,15,1,16
Gauss code	{1, -7, 2, -8, 5, -9}, {7, -1, 3, -4, 8, -2, 4, -6, 9, -5, 6, -3}

A Braid Representative	{{{braid_table}}}
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

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

-12

1

2

-1

-14

1

Integral Khovanov Homology

(db, data source)

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