L9a45

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	[edit L9a45 Quick Notes] L9a45 is $9_{7}^{3}$ 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_18,12,9,11 X_16,14,17,13 X_8,16,5,15 X_14,8,15,7 X_12,18,13,17 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -8, 2, -9}, {8, -1, 6, -5}, {9, -2, 3, -7, 4, -6, 5, -4, 7, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2t(2)t(1)+2t(2)t(3)t(1)-2t(3)t(1)+3t(1)+2t(2)-3t(2)t(3)+2t(3)-2}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}}}$ (db)
Jones polynomial	$-q^{5}+3q^{4}-4q^{3}+5q^{2}-6q+7-4q^{-1}+4q^{-2}-q^{-3}+q^{-4}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{-2}+a^{4}-2z^{2}a^{2}-2a^{2}z^{-2}-3a^{2}+z^{4}+z^{2}+z^{-2}+2+z^{4}a^{-2}+z^{2}a^{-2}-z^{2}a^{-4}$ (db)
Kauffman polynomial	$z^{5}a^{-5}-2z^{3}a^{-5}+3z^{6}a^{-4}+a^{4}z^{4}-8z^{4}a^{-4}-3a^{4}z^{2}+3z^{2}a^{-4}-a^{4}z^{-2}+3a^{4}+3z^{7}a^{-3}+a^{3}z^{5}-8z^{5}a^{-3}+5z^{3}a^{-3}-3a^{3}z+2a^{3}z^{-1}+z^{8}a^{-2}+a^{2}z^{6}+z^{6}a^{-2}+2a^{2}z^{4}-5z^{4}a^{-2}-6a^{2}z^{2}+2z^{2}a^{-2}-2a^{2}z^{-2}+5a^{2}+az^{7}+4z^{7}a^{-1}+az^{5}-9z^{5}a^{-1}+7z^{3}a^{-1}-3az+2az^{-1}+z^{8}-z^{6}+4z^{4}-4z^{2}-z^{-2}+3$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

2

7

2

1

-1

5

3

2

1

3

2

-1

1

4

3

1

-1

3

6

3

-3

1

0

-5

3

-7

1

0

-9

1

Integral Khovanov Homology

(db, data source)

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