L11a383

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{4}v^{4}-u^{4}v^{3}-u^{3}v^{4}+4u^{3}v^{3}-3u^{3}v^{2}-3u^{2}v^{3}+7u^{2}v^{2}-3u^{2}v-3uv^{2}+4uv-u-v+1}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-4q^{9/2}+{\frac {1}{q^{9/2}}}+7q^{7/2}-{\frac {2}{q^{7/2}}}-9q^{5/2}+{\frac {4}{q^{5/2}}}+10q^{3/2}-{\frac {7}{q^{3/2}}}-q^{13/2}+2q^{11/2}-11{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-3}-6z^{5}a^{-3}-12z^{3}a^{-3}-8za^{-3}-a^{-3}z^{-1}+z^{9}a^{-1}-az^{7}+8z^{7}a^{-1}-6az^{5}+24z^{5}a^{-1}-12az^{3}+32z^{3}a^{-1}-8az+17za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-3z^{3}a^{-7}+za^{-7}+2z^{6}a^{-6}-5z^{4}a^{-6}+z^{2}a^{-6}+3z^{7}a^{-5}-8z^{5}a^{-5}+5z^{3}a^{-5}-za^{-5}+4z^{8}a^{-4}+a^{4}z^{6}-15z^{6}a^{-4}-4a^{4}z^{4}+22z^{4}a^{-4}+4a^{4}z^{2}-10z^{2}a^{-4}+3z^{9}a^{-3}+2a^{3}z^{7}-11z^{7}a^{-3}-7a^{3}z^{5}+18z^{5}a^{-3}+6a^{3}z^{3}-11z^{3}a^{-3}-a^{3}z+7za^{-3}-a^{-3}z^{-1}+z^{10}a^{-2}+2a^{2}z^{8}+z^{8}a^{-2}-5a^{2}z^{6}-14z^{6}a^{-2}+2a^{2}z^{4}+31z^{4}a^{-2}-a^{2}z^{2}-16z^{2}a^{-2}+a^{-2}+2az^{9}+5z^{9}a^{-1}-7az^{7}-23z^{7}a^{-1}+15az^{5}+49z^{5}a^{-1}-22az^{3}-47z^{3}a^{-1}+9az+19za^{-1}-a^{-1}z^{-1}+z^{10}-z^{8}-3z^{6}+10z^{4}-10z^{2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

1

-1

10

3

1

2

8

4

1

-3

6

5

3

2

4

6

5

-1

2

5

4

1

0

4

7

3

-2

3

4

-1

-4

1

4

3

-6

1

3

-2

-8

1

-10

1

-1

Integral Khovanov Homology

(db, data source)

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