L11a250

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{5}-3u^{3}v^{4}+3u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{5}+4u^{2}v^{4}-8u^{2}v^{3}+5u^{2}v^{2}-u^{2}v-uv^{4}+5uv^{3}-8uv^{2}+4uv-u-v^{3}+3v^{2}-3v+1}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-q^{5/2}+4q^{3/2}-7{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {15}{q^{3/2}}}+{\frac {16}{q^{5/2}}}-{\frac {17}{q^{7/2}}}+{\frac {15}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{3}z^{9}-a^{5}z^{7}+6a^{3}z^{7}-az^{7}-4a^{5}z^{5}+11a^{3}z^{5}-4az^{5}-3a^{5}z^{3}+4a^{3}z^{3}-3az^{3}+2a^{5}z-5a^{3}z+az+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}+4a^{9}z^{5}-3a^{9}z^{3}+7a^{8}z^{6}-7a^{8}z^{4}+a^{8}z^{2}+8a^{7}z^{7}-7a^{7}z^{5}-3a^{7}z^{3}+2a^{7}z+8a^{6}z^{8}-10a^{6}z^{6}-a^{6}z^{4}+3a^{6}z^{2}+7a^{5}z^{9}-15a^{5}z^{7}+13a^{5}z^{5}-7a^{5}z^{3}-a^{5}z+a^{5}z^{-1}+3a^{4}z^{10}+a^{4}z^{8}-21a^{4}z^{6}+21a^{4}z^{4}-2a^{4}z^{2}-a^{4}+13a^{3}z^{9}-48a^{3}z^{7}+54a^{3}z^{5}-17a^{3}z^{3}-4a^{3}z+a^{3}z^{-1}+3a^{2}z^{10}-3a^{2}z^{8}-19a^{2}z^{6}+27a^{2}z^{4}-6a^{2}z^{2}+6az^{9}-24az^{7}+z^{7}a^{-1}+27az^{5}-3z^{5}a^{-1}-9az^{3}+z^{3}a^{-1}-az+4z^{8}-15z^{6}+13z^{4}-2z^{2}$ (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

6

1

4

3

-3

2

4

1

3

0

6

3

-3

-2

9

4

5

-4

8

7

-1

-6

9

8

1

-8

6

8

2

-10

5

9

-4

-12

3

7

4

-14

1

4

-3

-16

3

-18

1

-1

Integral Khovanov Homology

(db, data source)

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