L9a13

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	[edit L9a13 Quick Notes] L9a13 is $9_{16}^{2}$ 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_18,13,5,14 X_14,7,15,8 X_8,17,9,18 X_16,9,17,10 X_10,15,11,16 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 4, -5, 6, -7, 9, -2, 3, -4, 7, -6, 5, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)^{3}+4t(1)t(2)^{2}-5t(2)^{2}-5t(1)t(2)+4t(2)+2t(1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {8}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {4}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {1}{q^{21/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+3a^{9}z+2a^{9}z^{-1}-2a^{7}z^{3}-a^{7}z-3a^{5}z^{3}-4a^{5}z-a^{5}z^{-1}-a^{3}z^{3}$ (db)
Kauffman polynomial	$-z^{6}a^{12}+4z^{4}a^{12}-5z^{2}a^{12}+2a^{12}-2z^{7}a^{11}+7z^{5}a^{11}-7z^{3}a^{11}+2za^{11}-a^{11}z^{-1}-z^{8}a^{10}-2z^{6}a^{10}+15z^{4}a^{10}-17z^{2}a^{10}+5a^{10}-6z^{7}a^{9}+16z^{5}a^{9}-12z^{3}a^{9}+6za^{9}-2a^{9}z^{-1}-z^{8}a^{8}-7z^{6}a^{8}+21z^{4}a^{8}-13z^{2}a^{8}+3a^{8}-4z^{7}a^{7}+3z^{5}a^{7}+3z^{3}a^{7}-6z^{6}a^{6}+7z^{4}a^{6}-z^{2}a^{6}-a^{6}-6z^{5}a^{5}+7z^{3}a^{5}-4za^{5}+a^{5}z^{-1}-3z^{4}a^{4}-z^{3}a^{3}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

-4

3

1

-2

-6

3

-8

3

0

-10

5

3

2

-12

3

4

1

-14

3

4

-1

-16

1

3

2

-18

1

3

-2

-20

1

-22

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$