L9a47

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	[edit L9a47 Quick Notes] L9a47 is $9_{2}^{3}$ in the Rolfsen table of links.

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-t(1)t(3)t(2)^{2}+2t(3)t(2)^{2}-t(2)^{2}-t(1)t(3)^{2}t(2)+2t(3)^{2}t(2)-2t(1)t(2)+4t(1)t(3)t(2)-4t(3)t(2)+t(2)+t(1)t(3)^{2}-t(3)^{2}+t(1)-2t(1)t(3)+t(3)}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$-q^{-8}+3q^{-7}-5q^{-6}+8q^{-5}-8q^{-4}+10q^{-3}-7q^{-2}+q+6q^{-1}-3$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{8}+3z^{2}a^{6}+a^{6}z^{-2}+3a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-2a^{4}z^{-2}-5a^{4}-z^{4}a^{2}+z^{2}a^{2}+a^{2}z^{-2}+3a^{2}+z^{2}$ (db)
Kauffman polynomial	$z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-7z^{4}a^{8}+5z^{2}a^{8}-2a^{8}+3z^{7}a^{7}-3z^{5}a^{7}-4z^{3}a^{7}+3za^{7}+z^{8}a^{6}+8z^{6}a^{6}-24z^{4}a^{6}+21z^{2}a^{6}+a^{6}z^{-2}-8a^{6}+7z^{7}a^{5}-10z^{5}a^{5}+5za^{5}-2a^{5}z^{-1}+z^{8}a^{4}+10z^{6}a^{4}-26z^{4}a^{4}+24z^{2}a^{4}+2a^{4}z^{-2}-9a^{4}+4z^{7}a^{3}-3z^{5}a^{3}-z^{3}a^{3}+3za^{3}-2a^{3}z^{-1}+5z^{6}a^{2}-8z^{4}a^{2}+7z^{2}a^{2}+a^{2}z^{-2}-4a^{2}+3z^{5}a-3z^{3}a+z^{4}-z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

3

1

2

-2

-1

4

1

3

-3

4

3

-1

-5

6

3

-7

4

6

2

-9

4

0

-11

2

5

3

-13

1

3

-2

-15

2

-17

1

-1

Integral Khovanov Homology

(db, data source)

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