L9a43

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	[edit L9a43 Quick Notes] L9a43 is $9_{4}^{3}$ 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_16,11,17,12 X_18,15,9,16 X_12,17,13,18 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

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{3}+t(2)t(3)^{3}-t(3)^{3}-3t(1)t(3)^{2}+2t(1)t(2)t(3)^{2}-3t(2)t(3)^{2}+2t(3)^{2}+3t(1)t(3)-2t(1)t(2)t(3)+3t(2)t(3)-2t(3)-t(1)+t(1)t(2)-t(2)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{-11}+2q^{-10}-5q^{-9}+8q^{-8}-8q^{-7}+10q^{-6}-7q^{-5}+7q^{-4}-3q^{-3}+q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$-a^{12}z^{-2}+4a^{10}z^{-2}+4a^{10}-6a^{8}z^{2}-5a^{8}z^{-2}-11a^{8}+3a^{6}z^{4}+8a^{6}z^{2}+2a^{6}z^{-2}+7a^{6}+a^{4}z^{4}+a^{4}z^{2}$ (db)
Kauffman polynomial	$a^{13}z^{5}-3a^{13}z^{3}+3a^{13}z-a^{13}z^{-1}+2a^{12}z^{6}-4a^{12}z^{4}+3a^{12}z^{2}+a^{12}z^{-2}-2a^{12}+2a^{11}z^{7}+a^{11}z^{5}-11a^{11}z^{3}+13a^{11}z-5a^{11}z^{-1}+a^{10}z^{8}+6a^{10}z^{6}-16a^{10}z^{4}+14a^{10}z^{2}+4a^{10}z^{-2}-10a^{10}+6a^{9}z^{7}-3a^{9}z^{5}-16a^{9}z^{3}+21a^{9}z-9a^{9}z^{-1}+a^{8}z^{8}+10a^{8}z^{6}-24a^{8}z^{4}+23a^{8}z^{2}+5a^{8}z^{-2}-14a^{8}+4a^{7}z^{7}-10a^{7}z^{3}+11a^{7}z-5a^{7}z^{-1}+6a^{6}z^{6}-11a^{6}z^{4}+11a^{6}z^{2}+2a^{6}z^{-2}-7a^{6}+3a^{5}z^{5}-2a^{5}z^{3}+a^{4}z^{4}-a^{4}z^{2}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

1

-5

3

1

-2

-7

4

-9

3

0

-11

7

4

3

-13

4

6

2

-15

4

0

-17

1

4

3

-19

1

4

-3

-21

1

-23

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$
$r=-9$	${\mathbb {Z} }$
$r=-8$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$