L11a246

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{4}-3u^{3}v^{3}+3u^{3}v^{2}-u^{3}v+u^{2}v^{5}-5u^{2}v^{4}+9u^{2}v^{3}-10u^{2}v^{2}+5u^{2}v-u^{2}-uv^{5}+5uv^{4}-10uv^{3}+9uv^{2}-5uv+u-v^{4}+3v^{3}-3v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {15}{q^{9/2}}}-q^{7/2}+{\frac {21}{q^{7/2}}}+5q^{5/2}-{\frac {25}{q^{5/2}}}-11q^{3/2}+{\frac {25}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+17{\sqrt {q}}-{\frac {23}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+3a^{3}z^{5}+2az^{5}-z^{5}a^{-1}-2a^{5}z^{3}+3a^{3}z^{3}-az^{3}-z^{3}a^{-1}-3az+za^{-1}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+4a^{7}z^{7}-10a^{7}z^{5}+6a^{7}z^{3}+7a^{6}z^{8}-17a^{6}z^{6}+13a^{6}z^{4}-3a^{6}z^{2}+7a^{5}z^{9}-12a^{5}z^{7}+3a^{5}z^{5}+6a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+3a^{4}z^{10}+9a^{4}z^{8}-32a^{4}z^{6}+29a^{4}z^{4}-7a^{4}z^{2}-a^{4}+16a^{3}z^{9}-30a^{3}z^{7}+17a^{3}z^{5}+z^{5}a^{-3}-a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+3a^{2}z^{10}+15a^{2}z^{8}-36a^{2}z^{6}+5z^{6}a^{-2}+21a^{2}z^{4}-4z^{4}a^{-2}-4a^{2}z^{2}+9az^{9}-3az^{7}+11z^{7}a^{-1}-12az^{5}-15z^{5}a^{-1}+3az^{3}+4z^{3}a^{-1}+2az+za^{-1}+13z^{8}-17z^{6}+3z^{4}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

4

-4

4

7

1

6

2

10

4

-6

0

13

7

6

-2

13

11

-2

-4

12

0

-6

9

13

4

-8

6

12

-6

-10

3

10

7

-12

1

5

-4

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$