L9n25

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	[edit L9n25 Quick Notes] L9n25 is $9_{18}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₈₄₉₃ X_2,14,3,13 X_14,7,15,8 X_11,16,12,17 X_9,11,10,18 X_17,5,18,10 X_4,15,1,16
Gauss code	{1, -4, 3, -9}, {-2, -1, 5, -3, -7, 8}, {-6, 2, 4, -5, 9, 6, -8, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)(w-1)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}}}$ (db)
Jones polynomial	$-q^{-5}+2q^{-4}-2q^{-3}+q^{2}+3q^{-2}-q-2q^{-1}+4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{4}-a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}+3a^{2}-2z^{2}-2z^{-2}-3+a^{-2}z^{-2}+a^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{7}+az^{7}+2a^{4}z^{6}+3a^{2}z^{6}+z^{6}+a^{5}z^{5}-2a^{3}z^{5}-3az^{5}-7a^{4}z^{4}-11a^{2}z^{4}-4z^{4}-3a^{5}z^{3}-2a^{3}z^{3}+2az^{3}+z^{3}a^{-1}+5a^{4}z^{2}+13a^{2}z^{2}+z^{2}a^{-2}+9z^{2}+a^{5}z+3a^{3}z+3az+za^{-1}-2a^{4}-6a^{2}-2a^{-2}-5-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

χ

5

1

3

0

1

4

1

3

-1

2

4

2

-3

1

-5

1

2

1

-7

1

0

-9

1

-11

1

-1

Integral Khovanov Homology

(db, data source)

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