L9a50

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	[edit L9a50 Quick Notes] L9a50 is $9_{1}^{3}$ 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_18,16,11,15 X_14,8,15,7 X_10,12,5,11 X_8,17,9,18 X_16,9,17,10 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -6, 7, -5}, {5, -2, 9, -4, 3, -7, 6, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{2}t(2)^{2}-t(3)^{2}t(2)^{2}-t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-t(2)^{2}-2t(1)t(3)^{2}t(2)+t(3)^{2}t(2)-t(1)t(2)+2t(1)t(3)t(2)-2t(3)t(2)+2t(2)+t(1)t(3)^{2}+t(1)-2t(1)t(3)+t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-5q^{4}+7q^{3}+q^{-3}-7q^{2}-2q^{-2}+8q+5q^{-1}-5$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{4}a^{-4}-2z^{2}a^{-4}-a^{-4}+z^{6}a^{-2}+4z^{4}a^{-2}+a^{2}z^{2}+6z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}+2a^{2}+4a^{-2}-2z^{4}-6z^{2}-2z^{-2}-5$ (db)
Kauffman polynomial	$z^{8}a^{-2}+z^{8}+2az^{7}+6z^{7}a^{-1}+4z^{7}a^{-3}+a^{2}z^{6}+8z^{6}a^{-2}+6z^{6}a^{-4}+3z^{6}-6az^{5}-14z^{5}a^{-1}-3z^{5}a^{-3}+5z^{5}a^{-5}-4a^{2}z^{4}-27z^{4}a^{-2}-9z^{4}a^{-4}+3z^{4}a^{-6}-19z^{4}+3az^{3}+3z^{3}a^{-1}-5z^{3}a^{-3}-4z^{3}a^{-5}+z^{3}a^{-7}+6a^{2}z^{2}+23z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-6}+22z^{2}+3az+5za^{-1}+3za^{-3}+za^{-5}-4a^{2}-8a^{-2}-2a^{-4}-9-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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

3

1

-2

7

4

2

5

4

0

3

4

3

1

3

6

3

-1

2

0

-3

1

4

3

-5

1

-1

-7

1

Integral Khovanov Homology

(db, data source)

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