L9a2

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	[edit L9a2 Quick Notes] L9a2 is $9_{31}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_12,6,13,5 X₈₄₉₃ X_16,10,17,9 X_18,12,5,11 X_10,18,11,17 X_2,14,3,13
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -5, 6, -8, 7, -4, 9, -2, 3, -6, 8, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-t(2)^{3}+t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-4q^{11/2}+6q^{9/2}-7q^{7/2}+6q^{5/2}-6q^{3/2}+3{\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-z^{5}a^{-1}+5z^{5}a^{-3}-z^{5}a^{-5}-3z^{3}a^{-1}+7z^{3}a^{-3}-3z^{3}a^{-5}+za^{-3}-za^{-5}+2a^{-1}z^{-1}-3a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}-2z^{2}a^{-8}+4z^{5}a^{-7}-3z^{3}a^{-7}+4z^{6}a^{-6}-3z^{4}a^{-6}-2z^{2}a^{-6}+a^{-6}+4z^{7}a^{-5}-8z^{5}a^{-5}+6z^{3}a^{-5}-2za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-2z^{6}a^{-4}-3z^{4}a^{-4}+3a^{-4}+7z^{7}a^{-3}-24z^{5}a^{-3}+22z^{3}a^{-3}-3za^{-3}-3a^{-3}z^{-1}+2z^{8}a^{-2}-5z^{6}a^{-2}+z^{2}a^{-2}+3a^{-2}+3z^{7}a^{-1}-12z^{5}a^{-1}+12z^{3}a^{-1}-za^{-1}-2a^{-1}z^{-1}+z^{6}-3z^{4}+z^{2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

16

1

14

2

-2

12

2

1

10

4

2

-2

8

3

2

1

6

3

4

1

4

3

0

2

5

3

0

1

0

-2

2

-4

1

-1

Integral Khovanov Homology

(db, data source)

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