L9a11

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	[edit L9a11 Quick Notes] L9a11 is $9_{26}^{2}$ 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_14,8,15,7 X_18,16,5,15 X_16,9,17,10 X_8,17,9,18 X_10,14,11,13 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -6, 5, -7, 9, -2, 7, -3, 4, -5, 6, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

3

1

-2

0

5

3

2

-2

5

4

-1

-4

4

0

-6

3

5

2

-8

2

4

-2

-10

1

4

3

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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