L9a10

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	[edit L9a10 Quick Notes] L9a10 is $9_{36}^{2}$ in the Rolfsen table of links.

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2(u-1)(v-1)\left(v^{2}-v+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-q^{9/2}+{\frac {1}{q^{9/2}}}+3q^{7/2}-{\frac {3}{q^{7/2}}}-6q^{5/2}+{\frac {4}{q^{5/2}}}+7q^{3/2}-{\frac {7}{q^{3/2}}}-8{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}+z^{5}a^{-1}-a^{3}z^{3}+2az^{3}+2z^{3}a^{-1}-z^{3}a^{-3}-a^{3}z+2za^{-1}-za^{-3}+a^{3}z^{-1}-2az^{-1}+2a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-5}+a^{4}z^{6}-3a^{4}z^{4}+3z^{4}a^{-4}+2a^{4}z^{2}+3a^{3}z^{7}-11a^{3}z^{5}+6z^{5}a^{-3}+11a^{3}z^{3}-6z^{3}a^{-3}-a^{3}z+3za^{-3}-a^{3}z^{-1}-a^{-3}z^{-1}+2a^{2}z^{8}-3a^{2}z^{6}+7z^{6}a^{-2}-4a^{2}z^{4}-10z^{4}a^{-2}+4a^{2}z^{2}+4z^{2}a^{-2}+8az^{7}+5z^{7}a^{-1}-22az^{5}-5z^{5}a^{-1}+13az^{3}-5z^{3}a^{-1}+2az+6za^{-1}-2az^{-1}-2a^{-1}z^{-1}+2z^{8}+3z^{6}-14z^{4}+6z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

10

1

8

2

-2

6

4

1

3

4

3

2

-1

2

5

4

1

0

5

0

-2

2

3

-1

-4

2

5

3

-6

1

2

-1

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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