L11a374

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(uv-u+1)(uv-v+1)\left(u^{2}v^{2}+uv+1\right)}{u^{2}v^{2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {7}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {4}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}z^{7}+6a^{7}z^{5}+12a^{7}z^{3}+9a^{7}z+a^{7}z^{-1}-a^{5}z^{9}-8a^{5}z^{7}-24a^{5}z^{5}-33a^{5}z^{3}-18a^{5}z-a^{5}z^{-1}+a^{3}z^{7}+6a^{3}z^{5}+11a^{3}z^{3}+6a^{3}z$ (db)
Kauffman polynomial	$-z^{3}a^{13}+za^{13}-2z^{4}a^{12}+z^{2}a^{12}-3z^{5}a^{11}+2z^{3}a^{11}-za^{11}-4z^{6}a^{10}+6z^{4}a^{10}-4z^{2}a^{10}-4z^{7}a^{9}+7z^{5}a^{9}-2z^{3}a^{9}-4z^{8}a^{8}+11z^{6}a^{8}-9z^{4}a^{8}+5z^{2}a^{8}-3z^{9}a^{7}+10z^{7}a^{7}-12z^{5}a^{7}+15z^{3}a^{7}-9za^{7}+a^{7}z^{-1}-z^{10}a^{6}-z^{8}a^{6}+18z^{6}a^{6}-29z^{4}a^{6}+16z^{2}a^{6}-a^{6}-5z^{9}a^{5}+26z^{7}a^{5}-46z^{5}a^{5}+40z^{3}a^{5}-18za^{5}+a^{5}z^{-1}-z^{10}a^{4}+2z^{8}a^{4}+9z^{6}a^{4}-23z^{4}a^{4}+12z^{2}a^{4}-2z^{9}a^{3}+12z^{7}a^{3}-24z^{5}a^{3}+20z^{3}a^{3}-7za^{3}-z^{8}a^{2}+6z^{6}a^{2}-11z^{4}a^{2}+6z^{2}a^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

2

1

-1

0

1

-2

2

1

-1

-4

4

1

3

-6

3

0

-8

5

3

2

-10

3

0

-12

4

5

-1

-14

2

3

1

-16

2

4

-2

-18

1

3

2

-20

1

-1

-22

1

Integral Khovanov Homology

(db, data source)

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