L11a115

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2uv^{3}-7uv^{2}+10uv-4u-4v^{3}+10v^{2}-7v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {14}{q^{7/2}}}+q^{5/2}-{\frac {15}{q^{5/2}}}-4q^{3/2}+{\frac {13}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {5}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+7{\sqrt {q}}-{\frac {11}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-3a^{7}z-2a^{7}z^{-1}+3a^{5}z^{3}+3a^{5}z+a^{5}z^{-1}-a^{3}z^{5}+a^{3}z^{-1}-az^{5}-az^{3}+z^{3}a^{-1}-2az-az^{-1}$ (db)
Kauffman polynomial	$a^{9}z^{7}-5a^{9}z^{5}+8a^{9}z^{3}-5a^{9}z+a^{9}z^{-1}+2a^{8}z^{8}-8a^{8}z^{6}+9a^{8}z^{4}-4a^{8}z^{2}+a^{8}+2a^{7}z^{9}-4a^{7}z^{7}-7a^{7}z^{5}+18a^{7}z^{3}-12a^{7}z+2a^{7}z^{-1}+a^{6}z^{10}+3a^{6}z^{8}-20a^{6}z^{6}+25a^{6}z^{4}-12a^{6}z^{2}+3a^{6}+6a^{5}z^{9}-15a^{5}z^{7}+4a^{5}z^{5}+11a^{5}z^{3}-8a^{5}z+a^{5}z^{-1}+a^{4}z^{10}+8a^{4}z^{8}-29a^{4}z^{6}+30a^{4}z^{4}-12a^{4}z^{2}+2a^{4}+4a^{3}z^{9}-2a^{3}z^{7}-7a^{3}z^{5}+4a^{3}z^{3}+2a^{3}z-a^{3}z^{-1}+7a^{2}z^{8}-10a^{2}z^{6}+6a^{2}z^{4}+z^{4}a^{-2}-4a^{2}z^{2}+a^{2}+8az^{7}-9az^{5}+4z^{5}a^{-1}-3z^{3}a^{-1}+3az-az^{-1}+7z^{6}-7z^{4}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

4

1

-3

0

7

3

4

-2

7

5

-2

-4

8

6

2

-6

6

7

1

-8

5

8

-3

-10

4

7

3

-12

1

4

-3

-14

1

4

3

-16

1

-1

-18

1

Integral Khovanov Homology

(db, data source)

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