L9n22

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	[edit L9n22 Quick Notes] L9n22 is $9_{15}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_11,17,12,16 X_15,9,16,18 X_17,13,18,12 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 3, -4}, {9, -2, -5, 7, 4, -3, -6, 5, -7, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv+uw^{3}-uw^{2}+uw-u+vw^{3}-vw^{2}+vw-v-w^{3}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-2q+3-3q^{-1}+4q^{-2}-2q^{-3}+4q^{-4}-q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$2a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-5a^{4}z^{-2}-8a^{4}+2z^{4}a^{2}+8z^{2}a^{2}+4a^{2}z^{-2}+10a^{2}-2z^{2}-z^{-2}-3$ (db)
Kauffman polynomial	$a^{6}z^{6}-5a^{6}z^{4}+9a^{6}z^{2}+2a^{6}z^{-2}-7a^{6}+a^{5}z^{7}-2a^{5}z^{5}-5a^{5}z^{3}+11a^{5}z-5a^{5}z^{-1}+5a^{4}z^{6}-20a^{4}z^{4}+23a^{4}z^{2}+5a^{4}z^{-2}-14a^{4}+a^{3}z^{7}+2a^{3}z^{5}-17a^{3}z^{3}+21a^{3}z-9a^{3}z^{-1}+4a^{2}z^{6}-14a^{2}z^{4}+16a^{2}z^{2}+4a^{2}z^{-2}-10a^{2}+4az^{5}-12az^{3}+13az+3za^{-1}-5az^{-1}-a^{-1}z^{-1}+z^{4}+2z^{2}+z^{-2}-2$ (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		\

-6

-5

-4

-3

-2

-1

0

1

χ

3

2

-2

1

-1

3

0

-3

1

-5

1

3

2

-7

3

1

2

-9

1

4

3

-11

0

-13

1

Integral Khovanov Homology

(db, data source)

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