L9a20

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	[edit L9a20 Quick Notes] L9a20 is $9_{42}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9a20's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_16,11,17,12 X_10,4,11,3 X_2,15,3,16 X_12,5,13,6 X₆₇₁₈ X_14,10,15,9 X_18,14,7,13 X_4,18,5,17
Gauss code	{1, -4, 3, -9, 5, -6}, {6, -1, 7, -3, 2, -5, 8, -7, 4, -2, 9, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-3t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}-t(2)^{3}+3t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-3t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-7q^{3/2}+9{\sqrt {q}}-{\frac {12}{\sqrt {q}}}+{\frac {11}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-2a^{3}z^{3}+a^{3}z^{-1}+az^{7}+4az^{5}-z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}-az-az^{-1}$ (db)
Kauffman polynomial	$-3a^{2}z^{8}-3z^{8}-7a^{3}z^{7}-13az^{7}-6z^{7}a^{-1}-7a^{4}z^{6}-6a^{2}z^{6}-4z^{6}a^{-2}-3z^{6}-4a^{5}z^{5}+8a^{3}z^{5}+25az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}-a^{6}z^{4}+8a^{4}z^{4}+16a^{2}z^{4}+7z^{4}a^{-2}+14z^{4}+3a^{5}z^{3}-2a^{3}z^{3}-12az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}-2a^{4}z^{2}-6a^{2}z^{2}-2z^{2}a^{-2}-6z^{2}-a^{3}z-az-a^{2}+a^{3}z^{-1}+az^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

1

3

2

5

3

-2

0

7

4

3

-2

5

6

1

-4

5

6

-1

-6

3

6

3

-8

1

4

-3

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

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