L11a353

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Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_16,6,17,5 X_22,8,11,7 X_20,15,21,16 X_14,21,15,22 X_6,14,7,13 X_4,20,5,19 X_18,9,19,10 X_2,11,3,12 X_10,17,1,18
Gauss code	{1, -10, 2, -8, 3, -7, 4, -2, 9, -11}, {10, -1, 7, -6, 5, -3, 11, -9, 8, -5, 6, -4}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{2}-3u^{2}v+2u^{2}-uv^{2}+5uv-u+2v^{2}-3v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{11/2}-4q^{9/2}+10q^{7/2}-17q^{5/2}+21q^{3/2}-25{\sqrt {q}}+{\frac {24}{\sqrt {q}}}-{\frac {21}{q^{3/2}}}+{\frac {15}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+3az^{5}-4z^{5}a^{-1}+z^{5}a^{-3}-3a^{3}z^{3}+7az^{3}-9z^{3}a^{-1}+2z^{3}a^{-3}+a^{5}z-3a^{3}z+6az-7za^{-1}+3za^{-3}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}+a^{5}z^{7}-3a^{5}z^{5}+4z^{5}a^{-5}+3a^{5}z^{3}-a^{5}z+4a^{4}z^{8}-13a^{4}z^{6}+10z^{6}a^{-4}+14a^{4}z^{4}-7z^{4}a^{-4}-5a^{4}z^{2}+3z^{2}a^{-4}+6a^{3}z^{9}-16a^{3}z^{7}+17z^{7}a^{-3}+9a^{3}z^{5}-25z^{5}a^{-3}+4a^{3}z^{3}+16z^{3}a^{-3}-2a^{3}z-5za^{-3}+3a^{2}z^{10}+7a^{2}z^{8}+18z^{8}a^{-2}-42a^{2}z^{6}-29z^{6}a^{-2}+43a^{2}z^{4}+12z^{4}a^{-2}-11a^{2}z^{2}-z^{2}a^{-2}+17az^{9}+11z^{9}a^{-1}-38az^{7}-4z^{7}a^{-1}+9az^{5}-32z^{5}a^{-1}+14az^{3}+29z^{3}a^{-1}-6az-10za^{-1}+az^{-1}+a^{-1}z^{-1}+3z^{10}+21z^{8}-68z^{6}+49z^{4}-10z^{2}-1$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

3

8

7

1

-6

6

10

3

7

4

11

7

-4

2

14

10

4

0

12

13

1

-2

9

12

-3

-4

6

12

6

-6

3

9

-6

-8

1

6

5

-10

3

-3

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$