L9a51

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	[edit L9a51 Quick Notes] L9a51 is $9_{11}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_8,12,9,11 X_18,8,11,7 X_16,13,17,14 X_14,6,15,5 X_10,16,5,15 X_2,9,3,10 X_4,18,1,17
Gauss code	{1, -8, 2, -9}, {6, -1, 4, -3, 8, -7}, {3, -2, 5, -6, 7, -5, 9, -4}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{2}w^{2}-2uv^{2}w-2uvw^{2}+4uvw-2uv+uw^{2}-2uw+u-v^{2}w^{2}+2v^{2}w-v^{2}+2vw^{2}-4vw+2v+2w-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{7}-3q^{6}+7q^{5}-8q^{4}+11q^{3}-10q^{2}-q^{-2}+9q+4q^{-1}-6$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{2}a^{-6}+a^{-6}z^{-2}+a^{-6}-2z^{4}a^{-4}-4z^{2}a^{-4}-2a^{-4}z^{-2}-3a^{-4}+z^{6}a^{-2}+3z^{4}a^{-2}+3z^{2}a^{-2}+a^{-2}z^{-2}+a^{-2}-z^{4}-z^{2}+1$ (db)
Kauffman polynomial	$z^{4}a^{-8}-z^{2}a^{-8}+3z^{5}a^{-7}-2z^{3}a^{-7}+6z^{6}a^{-6}-10z^{4}a^{-6}+10z^{2}a^{-6}+a^{-6}z^{-2}-5a^{-6}+5z^{7}a^{-5}-3z^{5}a^{-5}-4z^{3}a^{-5}+6za^{-5}-2a^{-5}z^{-1}+2z^{8}a^{-4}+8z^{6}a^{-4}-21z^{4}a^{-4}+17z^{2}a^{-4}+2a^{-4}z^{-2}-8a^{-4}+10z^{7}a^{-3}-16z^{5}a^{-3}+2z^{3}a^{-3}+6za^{-3}-2a^{-3}z^{-1}+2z^{8}a^{-2}+6z^{6}a^{-2}-18z^{4}a^{-2}+9z^{2}a^{-2}+a^{-2}z^{-2}-3a^{-2}+5z^{7}a^{-1}+az^{5}-9z^{5}a^{-1}-az^{3}+3z^{3}a^{-1}+4z^{6}-8z^{4}+3z^{2}+1$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

15

1

13

3

1

-2

11

4

9

4

3

-1

7

4

3

5

4

5

1

3

5

6

-1

1

3

6

3

-1

1

3

-2

-3

3

-5

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{5}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=5$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }$	${\mathbb {Z} }$