L11a290

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{4}-3u^{3}v^{3}+3u^{3}v^{2}-u^{3}v+u^{2}v^{5}-4u^{2}v^{4}+8u^{2}v^{3}-9u^{2}v^{2}+4u^{2}v-u^{2}-uv^{5}+4uv^{4}-9uv^{3}+8uv^{2}-4uv+u-v^{4}+3v^{3}-3v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+15{\sqrt {q}}-{\frac {20}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {23}{q^{5/2}}}+{\frac {19}{q^{7/2}}}-{\frac {14}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+3a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-2a^{5}z^{3}+2a^{3}z^{3}+3az^{3}-2z^{3}a^{-1}-2a^{3}z+az-za^{-1}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+4a^{7}z^{7}-10a^{7}z^{5}+5a^{7}z^{3}+7a^{6}z^{8}-18a^{6}z^{6}+13a^{6}z^{4}-3a^{6}z^{2}+7a^{5}z^{9}-15a^{5}z^{7}+9a^{5}z^{5}-a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+3a^{4}z^{10}+6a^{4}z^{8}-29a^{4}z^{6}+30a^{4}z^{4}-8a^{4}z^{2}-a^{4}+15a^{3}z^{9}-37a^{3}z^{7}+36a^{3}z^{5}+z^{5}a^{-3}-13a^{3}z^{3}-z^{3}a^{-3}-a^{3}z+a^{3}z^{-1}+3a^{2}z^{10}+9a^{2}z^{8}-30a^{2}z^{6}+4z^{6}a^{-2}+29a^{2}z^{4}-5z^{4}a^{-2}-9a^{2}z^{2}+z^{2}a^{-2}+8az^{9}-10az^{7}+8z^{7}a^{-1}+4az^{5}-12z^{5}a^{-1}-az^{3}+5z^{3}a^{-1}-za^{-1}+10z^{8}-16z^{6}+9z^{4}-3z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

9

3

-6

0

11

6

5

-2

12

10

-2

-4

11

10

1

-6

8

12

4

-8

6

11

-5

-10

3

9

6

-12

1

5

-4

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$