L9n21

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	[edit L9n21 Quick Notes] L9n21 is $9_{17}^{3}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_16,11,17,12 X_18,15,9,16 X_12,17,13,18 X₂₅₃₆ X_9,1,10,4
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 $u$ , $v$ , $w$ , ...)	${\frac {-uv-uw^{2}+uw-vw^{2}+vw+w^{3}}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{-5}+q^{-4}+q^{3}-q^{-3}+q^{-2}+q+2$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-z^{2}a^{4}-a^{4}z^{-2}-2a^{4}+z^{4}a^{2}+5z^{2}a^{2}+4a^{2}z^{-2}+8a^{2}-z^{4}-6z^{2}-5z^{-2}-9+z^{2}a^{-2}+2a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$a^{4}z^{6}+a^{2}z^{6}+z^{6}a^{-2}+z^{6}+a^{5}z^{5}+2a^{3}z^{5}+2az^{5}+z^{5}a^{-1}-4a^{4}z^{4}-6a^{2}z^{4}-6z^{4}a^{-2}-8z^{4}-4a^{5}z^{3}-10a^{3}z^{3}-13az^{3}-7z^{3}a^{-1}+3a^{4}z^{2}+11a^{2}z^{2}+10z^{2}a^{-2}+18z^{2}+3a^{5}z+13a^{3}z+21az+11za^{-1}-2a^{4}-10a^{2}-7a^{-2}-14-a^{5}z^{-1}-5a^{3}z^{-1}-9az^{-1}-5a^{-1}z^{-1}+a^{4}z^{-2}+4a^{2}z^{-2}+2a^{-2}z^{-2}+5z^{-2}$ (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

3

4

χ

7

1

5

1

3

1

3

-1

1

4

1

2

-3

1

-5

1

0

-7

1

0

-9

0

-11

1

-1

Integral Khovanov Homology

(db, data source)

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