L9a55

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	[edit L9a55 Quick Notes] L9a55 is $9_{1}^{4}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L9a55's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_16,12,17,11 X_10,3,11,4 X_4,9,1,10 X_14,7,15,8 X_8,13,5,14 X_18,16,13,15 X_12,18,9,17
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, 3, -9}, {7, -6, 8, -3, 9, -8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(2)t(1)-t(2)t(3)t(1)+t(3)t(1)-t(2)t(4)t(1)+t(2)t(3)t(4)t(1)-2t(3)t(4)t(1)+2t(4)t(1)-2t(1)-2t(2)+2t(2)t(3)-t(3)+t(2)t(4)-2t(2)t(3)t(4)+2t(3)t(4)-t(4)+1}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}}}$ (db)
Jones polynomial	$q^{5/2}-4q^{3/2}+6{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {1}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}+a^{7}z^{-3}-3za^{5}-5a^{5}z^{-1}-3a^{5}z^{-3}+3z^{3}a^{3}+6za^{3}+7a^{3}z^{-1}+3a^{3}z^{-3}-z^{5}a-2z^{3}a-3za-3az^{-1}-az^{-3}+z^{3}a^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-a^{5}z^{7}-5a^{3}z^{7}-4az^{7}-a^{6}z^{6}-a^{4}z^{6}-6a^{2}z^{6}-6z^{6}-a^{7}z^{5}-3a^{5}z^{5}+4a^{3}z^{5}+2az^{5}-4z^{5}a^{-1}-2a^{4}z^{4}+7a^{2}z^{4}-z^{4}a^{-2}+8z^{4}+4a^{7}z^{3}+12a^{5}z^{3}+8a^{3}z^{3}+4az^{3}+4z^{3}a^{-1}+6a^{6}z^{2}+12a^{4}z^{2}+6a^{2}z^{2}-6a^{7}z-14a^{5}z-14a^{3}z-6az-8a^{6}-15a^{4}-8a^{2}+4a^{7}z^{-1}+9a^{5}z^{-1}+9a^{3}z^{-1}+4az^{-1}+3a^{6}z^{-2}+6a^{4}z^{-2}+3a^{2}z^{-2}-a^{7}z^{-3}-3a^{5}z^{-3}-3a^{3}z^{-3}-az^{-3}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

3

1

-2

0

5

3

2

-2

6

7

1

-4

4

1

3

-6

1

6

5

-8

4

0

-10

1

5

4

-12

0

-14

1

Integral Khovanov Homology

(db, data source)

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