L9n20

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	[edit L9n20 Quick Notes] L9n20 is $9_{16}^{3}$ 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_11,17,12,16 X_15,9,16,18 X_17,13,18,12 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 $u$ , $v$ , $w$ , ...)	${\frac {-2uvw^{2}+2uvw-uv+uw^{2}-uw+vw^{2}-vw+w^{3}-2w^{2}+2w}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$-q^{8}+2q^{7}-4q^{6}+5q^{5}-4q^{4}+6q^{3}-3q^{2}+3q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2z^{4}a^{-4}+3z^{2}a^{-2}-7z^{2}a^{-4}+3z^{2}a^{-6}+5a^{-2}-10a^{-4}+6a^{-6}-a^{-8}+2a^{-2}z^{-2}-5a^{-4}z^{-2}+4a^{-6}z^{-2}-a^{-8}z^{-2}$ (db)
Kauffman polynomial	$z^{7}a^{-5}+z^{7}a^{-7}+4z^{6}a^{-4}+6z^{6}a^{-6}+2z^{6}a^{-8}+3z^{5}a^{-3}+5z^{5}a^{-5}+3z^{5}a^{-7}+z^{5}a^{-9}-11z^{4}a^{-4}-16z^{4}a^{-6}-5z^{4}a^{-8}-6z^{3}a^{-3}-18z^{3}a^{-5}-15z^{3}a^{-7}-3z^{3}a^{-9}+6z^{2}a^{-2}+18z^{2}a^{-4}+15z^{2}a^{-6}+3z^{2}a^{-8}+11za^{-3}+21za^{-5}+13za^{-7}+3za^{-9}-7a^{-2}-14a^{-4}-10a^{-6}-2a^{-8}-5a^{-3}z^{-1}-9a^{-5}z^{-1}-5a^{-7}z^{-1}-a^{-9}z^{-1}+2a^{-2}z^{-2}+5a^{-4}z^{-2}+4a^{-6}z^{-2}+a^{-8}z^{-2}$ (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		\

0

1

2

3

4

5

6

7

χ

17

1

-1

15

1

13

3

1

-2

11

2

1

9

3

4

1

7

3

1

2

5

3

0

1

3

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$
$r=0$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }^{3}$
$r=1$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$