L9a8

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	[edit L9a8 Quick Notes] L9a8 is $9_{25}^{2}$ in the Rolfsen table of links.

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

3

1

-2

6

5

2

3

4

3

0

2

5

0

4

5

1

-2

1

3

-2

-4

1

4

3

-6

1

-1

-8

1

Integral Khovanov Homology

(db, data source)

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