L11a307

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{3}-3u^{3}v^{2}+3u^{3}v-3u^{2}v^{3}+9u^{2}v^{2}-8u^{2}v+3u^{2}+3uv^{3}-8uv^{2}+9uv-3u+3v^{2}-3v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {7}{\sqrt {q}}}-{\frac {12}{q^{3/2}}}+{\frac {17}{q^{5/2}}}-{\frac {19}{q^{7/2}}}+{\frac {19}{q^{9/2}}}-{\frac {17}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {8}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-3z^{3}a^{7}-5za^{7}-a^{7}z^{-1}+3z^{5}a^{5}+8z^{3}a^{5}+5za^{5}-z^{7}a^{3}-4z^{5}a^{3}-7z^{3}a^{3}-5za^{3}+z^{5}a+2z^{3}a$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-4a^{10}z^{4}+a^{10}z^{2}+6a^{9}z^{7}-11a^{9}z^{5}+11a^{9}z^{3}-7a^{9}z+a^{9}z^{-1}+6a^{8}z^{8}-5a^{8}z^{6}-3a^{8}z^{4}+4a^{8}z^{2}-a^{8}+4a^{7}z^{9}+3a^{7}z^{7}-14a^{7}z^{5}+12a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+a^{6}z^{10}+13a^{6}z^{8}-31a^{6}z^{6}+27a^{6}z^{4}-9a^{6}z^{2}+8a^{5}z^{9}-8a^{5}z^{7}-8a^{5}z^{5}+10a^{5}z^{3}-a^{5}z+a^{4}z^{10}+13a^{4}z^{8}-40a^{4}z^{6}+41a^{4}z^{4}-16a^{4}z^{2}+4a^{3}z^{9}-a^{3}z^{7}-17a^{3}z^{5}+18a^{3}z^{3}-4a^{3}z+6a^{2}z^{8}-16a^{2}z^{6}+13a^{2}z^{4}-4a^{2}z^{2}+4az^{7}-11az^{5}+7az^{3}+z^{6}-2z^{4}$ (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

4

1

-1

2

3

0

4

1

-3

-2

8

3

5

-4

10

5

-5

-6

9

7

2

-8

10

0

-10

7

9

-2

-12

5

10

5

-14

3

7

-4

-16

5

-18

1

3

-2

-20

1

Integral Khovanov Homology

(db, data source)

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