L11a282

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{3}t(2)^{5}-t(1)^{2}t(2)^{5}-2t(1)^{3}t(2)^{4}+4t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+2t(1)^{3}t(2)^{3}-6t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-t(2)^{3}-t(1)^{3}t(2)^{2}+5t(1)^{2}t(2)^{2}-6t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+4t(1)t(2)-2t(2)-t(1)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{13/2}+3q^{11/2}-6q^{9/2}+9q^{7/2}-13q^{5/2}+14q^{3/2}-15{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {6}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{9}a^{-1}-az^{7}+7z^{7}a^{-1}-z^{7}a^{-3}-5az^{5}+18z^{5}a^{-1}-5z^{5}a^{-3}-8az^{3}+21z^{3}a^{-1}-8z^{3}a^{-3}-5az+12za^{-1}-5za^{-3}-az^{-1}+3a^{-1}z^{-1}-2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-5az^{9}-9z^{9}a^{-1}-4z^{9}a^{-3}-5a^{2}z^{8}+z^{8}a^{-2}-4z^{8}a^{-4}-3a^{3}z^{7}+18az^{7}+35z^{7}a^{-1}+10z^{7}a^{-3}-4z^{7}a^{-5}-a^{4}z^{6}+17a^{2}z^{6}+7z^{6}a^{-2}+5z^{6}a^{-4}-3z^{6}a^{-6}+17z^{6}+9a^{3}z^{5}-24az^{5}-57z^{5}a^{-1}-17z^{5}a^{-3}+6z^{5}a^{-5}-z^{5}a^{-7}+3a^{4}z^{4}-16a^{2}z^{4}-18z^{4}a^{-2}+6z^{4}a^{-6}-31z^{4}-4a^{3}z^{3}+17az^{3}+39z^{3}a^{-1}+16z^{3}a^{-3}+2z^{3}a^{-7}-a^{4}z^{2}+7a^{2}z^{2}+11z^{2}a^{-2}-2z^{2}a^{-6}+17z^{2}-6az-15za^{-1}-8za^{-3}-za^{-7}-a^{2}-3a^{-2}-3+az^{-1}+3a^{-1}z^{-1}+2a^{-3}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

2

-2

10

4

1

3

8

6

3

-3

6

7

3

4

7

6

-1

2

8

7

1

0

5

8

3

-2

4

7

-3

-4

2

5

3

-6

1

4

-3

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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