L11a288

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_16,9,17,10 X_20,12,21,11 X_22,15,9,16 X_14,6,15,5 X_4,14,5,13 X_6,20,7,19 X_18,8,19,7 X_8,18,1,17 X_2,21,3,22
Gauss code	{1, -11, 2, -7, 6, -8, 9, -10}, {3, -1, 4, -2, 7, -6, 5, -3, 10, -9, 8, -4, 11, -5}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u+v)\left(uv^{2}-uv+u+v-1\right)\left(uv^{2}-uv-v^{2}+v-1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{15/2}-3q^{13/2}+6q^{11/2}-11q^{9/2}+13q^{7/2}-16q^{5/2}+16q^{3/2}-14{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{7/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+3z^{3}a^{-5}+2za^{-5}-a^{-5}z^{-1}-z^{7}a^{-3}-4z^{5}a^{-3}-5z^{3}a^{-3}-za^{-3}+a^{-3}z^{-1}-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+3az^{3}-5z^{3}a^{-1}+2az-za^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}a^{-4}-5z^{9}a^{-1}-9z^{9}a^{-3}-4z^{9}a^{-5}-2z^{8}a^{-2}-4z^{8}a^{-6}-6z^{8}-5az^{7}+12z^{7}a^{-1}+29z^{7}a^{-3}+9z^{7}a^{-5}-3z^{7}a^{-7}-3a^{2}z^{6}+16z^{6}a^{-2}+8z^{6}a^{-4}+9z^{6}a^{-6}-z^{6}a^{-8}+15z^{6}-a^{3}z^{5}+11az^{5}-13z^{5}a^{-1}-40z^{5}a^{-3}-6z^{5}a^{-5}+9z^{5}a^{-7}+6a^{2}z^{4}-26z^{4}a^{-2}-9z^{4}a^{-4}-2z^{4}a^{-6}+3z^{4}a^{-8}-16z^{4}+2a^{3}z^{3}-7az^{3}+2z^{3}a^{-1}+21z^{3}a^{-3}+3z^{3}a^{-5}-7z^{3}a^{-7}-a^{2}z^{2}+9z^{2}a^{-2}+3z^{2}a^{-4}-z^{2}a^{-6}-2z^{2}a^{-8}+6z^{2}+2az-za^{-1}-2za^{-3}+3za^{-5}+2za^{-7}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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

16

1

-1

14

2

12

4

1

-3

10

7

2

5

8

7

5

-2

6

9

6

3

4

7

0

2

7

9

-2

0

4

8

4

-2

2

6

-4

1

4

3

-6

2

-2

-8

1

Integral Khovanov Homology

(db, data source)

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