L9a44

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	[edit L9a44 Quick Notes] L9a44 is $9_{3}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_18,14,9,13 X_16,12,17,11 X_12,18,13,17 X_8,16,5,15 X_14,8,15,7 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 7, -6}, {9, -2, 4, -5, 3, -7, 6, -4, 5, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(3)^{3}+t(1)t(2)t(3)^{3}-2t(2)t(3)^{3}+t(3)^{3}+2t(1)t(3)^{2}-t(1)t(2)t(3)^{2}+2t(2)t(3)^{2}-t(3)^{2}-2t(1)t(3)+t(1)t(2)t(3)-2t(2)t(3)+t(3)+2t(1)-t(1)t(2)+t(2)-1}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-6q^{4}+7q^{3}-7q^{2}+8q-5+5q^{-1}-q^{-2}+q^{-3}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}a^{-2}+4z^{4}a^{-2}-z^{4}a^{-4}-2z^{4}+a^{2}z^{2}+7z^{2}a^{-2}-2z^{2}a^{-4}-7z^{2}+3a^{2}+8a^{-2}-2a^{-4}-9+2a^{2}z^{-2}+4a^{-2}z^{-2}-a^{-4}z^{-2}-5z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-7}+3z^{4}a^{-6}+6z^{5}a^{-5}-6z^{3}a^{-5}+3za^{-5}-a^{-5}z^{-1}+7z^{6}a^{-4}-11z^{4}a^{-4}+6z^{2}a^{-4}+a^{-4}z^{-2}-2a^{-4}+4z^{7}a^{-3}-z^{5}a^{-3}-12z^{3}a^{-3}+13za^{-3}-5a^{-3}z^{-1}+z^{8}a^{-2}+a^{2}z^{6}+7z^{6}a^{-2}-5a^{2}z^{4}-21z^{4}a^{-2}+9a^{2}z^{2}+16z^{2}a^{-2}+2a^{2}z^{-2}+4a^{-2}z^{-2}-7a^{2}-10a^{-2}+az^{7}+5z^{7}a^{-1}-az^{5}-8z^{5}a^{-1}-6az^{3}-11z^{3}a^{-1}+11az+21za^{-1}-5az^{-1}-9a^{-1}z^{-1}+z^{8}+z^{6}-12z^{4}+19z^{2}+5z^{-2}-14$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

13

1

-1

11

2

9

4

1

-3

7

3

2

1

5

4

0

3

4

3

1

4

7

3

-1

1

0

-3

4

-5

1

0

-7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{4}$
$r=1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$