L9a1

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	[edit L9a1 Quick Notes] L9a1 is $9_{32}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_10,6,11,5 X₈₄₉₃ X_18,12,5,11 X_12,18,13,17 X_16,10,17,9 X_2,14,3,13
Gauss code	{1, -9, 5, -3}, {4, -1, 2, -5, 8, -4, 6, -7, 9, -2, 3, -8, 7, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-5q^{9/2}+8q^{7/2}-10q^{5/2}+9q^{3/2}-9{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {4}{q^{3/2}}}+{\frac {1}{q^{5/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-1}+z^{5}a^{-3}-az^{3}+z^{3}a^{-1}+2z^{3}a^{-3}-z^{3}a^{-5}-za^{-1}+2za^{-3}-za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{8}a^{-2}-2z^{8}a^{-4}-5z^{7}a^{-1}-9z^{7}a^{-3}-4z^{7}a^{-5}-5z^{6}a^{-2}-2z^{6}a^{-4}-3z^{6}a^{-6}-6z^{6}-4az^{5}+4z^{5}a^{-1}+18z^{5}a^{-3}+9z^{5}a^{-5}-z^{5}a^{-7}-a^{2}z^{4}+11z^{4}a^{-2}+10z^{4}a^{-4}+7z^{4}a^{-6}+7z^{4}+4az^{3}-13z^{3}a^{-3}-7z^{3}a^{-5}+2z^{3}a^{-7}-4z^{2}a^{-2}-7z^{2}a^{-4}-4z^{2}a^{-6}-z^{2}+2za^{-1}+4za^{-3}+2za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

2

-2

10

3

1

2

8

5

2

-3

6

5

3

2

4

5

1

2

5

0

3

6

3

-2

1

3

-2

-4

3

-6

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
$r=0$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{5}$
$r=1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$