L11a90

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_18,8,19,7 X_20,10,21,9 X_22,15,5,16 X_8,20,9,19 X_16,21,17,22 X_14,12,15,11 X_10,18,11,17 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 4, -9, 8, -2, 11, -8, 5, -7, 9, -3, 6, -4, 7, -5}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{4}-4t(2)^{3}+3t(2)^{2}-4t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{17/2}-4q^{15/2}+8q^{13/2}-12q^{11/2}+15q^{9/2}-17q^{7/2}+16q^{5/2}-13q^{3/2}+9{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+4z^{5}a^{-3}-2z^{5}a^{-5}+az^{3}-7z^{3}a^{-1}+7z^{3}a^{-3}-5z^{3}a^{-5}+z^{3}a^{-7}+3az-8za^{-1}+7za^{-3}-3za^{-5}+za^{-7}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-2z^{9}a^{-1}-6z^{9}a^{-3}-4z^{9}a^{-5}-4z^{8}a^{-2}-10z^{8}a^{-4}-8z^{8}a^{-6}-2z^{8}-az^{7}+2z^{7}a^{-1}+8z^{7}a^{-3}-5z^{7}a^{-5}-10z^{7}a^{-7}+17z^{6}a^{-2}+25z^{6}a^{-4}+7z^{6}a^{-6}-8z^{6}a^{-8}+7z^{6}+5az^{5}+13z^{5}a^{-1}+14z^{5}a^{-3}+23z^{5}a^{-5}+13z^{5}a^{-7}-4z^{5}a^{-9}-9z^{4}a^{-2}-9z^{4}a^{-4}+3z^{4}a^{-6}+8z^{4}a^{-8}-z^{4}a^{-10}-6z^{4}-9az^{3}-26z^{3}a^{-1}-24z^{3}a^{-3}-15z^{3}a^{-5}-6z^{3}a^{-7}+2z^{3}a^{-9}-7z^{2}a^{-2}-6z^{2}a^{-4}-2z^{2}a^{-6}-2z^{2}a^{-8}-z^{2}+7az+17za^{-1}+13za^{-3}+4za^{-5}+za^{-7}+3a^{-2}+3a^{-4}+a^{-6}+2-2az^{-1}-4a^{-1}z^{-1}-3a^{-3}z^{-1}-a^{-5}z^{-1}$ (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

6

7

χ

18

1

-1

16

3

14

5

1

-4

12

7

3

4

10

8

5

-3

8

9

7

2

6

7

8

1

4

6

9

-3

2

5

9

4

0

1

4

-3

-2

1

5

4

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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