L9a42

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	[edit L9a42 Quick Notes] L9a42 is $9_{41}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_18,5,9,6 X_6,9,7,10 X_16,12,17,11 X_14,8,15,7 X_4,14,5,13 X_8,16,1,15 X_2,17,3,18
Gauss code	{1, -9, 2, -7, 3, -4, 6, -8}, {4, -1, 5, -2, 7, -6, 8, -5, 9, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{2}-uv^{2}+3uv-u+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+6q^{7/2}-{\frac {1}{q^{7/2}}}-9q^{5/2}+{\frac {3}{q^{5/2}}}+9q^{3/2}-{\frac {6}{q^{3/2}}}+q^{11/2}-10{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-5z^{5}a^{-1}+z^{5}a^{-3}+3az^{3}-9z^{3}a^{-1}+3z^{3}a^{-3}+3az-6za^{-1}+3za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+3z^{5}a^{-5}-3z^{3}a^{-5}+za^{-5}+5z^{6}a^{-4}-6z^{4}a^{-4}+3z^{2}a^{-4}+5z^{7}a^{-3}+a^{3}z^{5}-6z^{5}a^{-3}-2a^{3}z^{3}+4z^{3}a^{-3}+a^{3}z-2za^{-3}+2z^{8}a^{-2}+3a^{2}z^{6}+5z^{6}a^{-2}-6a^{2}z^{4}-12z^{4}a^{-2}+3a^{2}z^{2}+6z^{2}a^{-2}+4az^{7}+9z^{7}a^{-1}-7az^{5}-17z^{5}a^{-1}+3az^{3}+12z^{3}a^{-1}-2az-6za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{8}+3z^{6}-11z^{4}+5z^{2}-1$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

4

1

-3

6

5

2

3

4

0

2

6

5

1

0

4

6

2

-2

2

4

-2

-4

1

4

3

-6

2

-2

-8

1

Integral Khovanov Homology

(db, data source)

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