L11a367

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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_18,7,19,8 X_8,11,9,12 X_20,10,21,9 X_22,20,11,19 X_10,22,1,21 X_4,15,5,16 X_6,17,7,18
Gauss code	{1, -2, 3, -10, 4, -11, 5, -6, 7, -9}, {6, -1, 2, -3, 10, -4, 11, -5, 8, -7, 9, -8}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{4}v^{4}-u^{4}v^{3}-u^{3}v^{4}+3u^{3}v^{3}-u^{3}v^{2}-u^{2}v^{3}+3u^{2}v^{2}-u^{2}v-uv^{2}+3uv-u-v+1}{u^{2}v^{2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {4}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {5}{q^{13/2}}}+{\frac {4}{q^{15/2}}}-{\frac {3}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}z^{7}+6a^{7}z^{5}+11a^{7}z^{3}+7a^{7}z+a^{7}z^{-1}-a^{5}z^{9}-8a^{5}z^{7}-23a^{5}z^{5}-29a^{5}z^{3}-14a^{5}z-a^{5}z^{-1}+a^{3}z^{7}+6a^{3}z^{5}+10a^{3}z^{3}+4a^{3}z$ (db)
Kauffman polynomial	$a^{13}z^{3}-a^{13}z+2a^{12}z^{4}-2a^{12}z^{2}+2a^{11}z^{5}-a^{11}z^{3}+2a^{10}z^{6}-2a^{10}z^{4}+a^{10}z^{2}+2a^{9}z^{7}-4a^{9}z^{5}+3a^{9}z^{3}-a^{9}z+2a^{8}z^{8}-6a^{8}z^{6}+5a^{8}z^{4}-2a^{8}z^{2}+2a^{7}z^{9}-9a^{7}z^{7}+15a^{7}z^{5}-16a^{7}z^{3}+8a^{7}z-a^{7}z^{-1}+a^{6}z^{10}-3a^{6}z^{8}-2a^{6}z^{6}+9a^{6}z^{4}-7a^{6}z^{2}+a^{6}+4a^{5}z^{9}-24a^{5}z^{7}+48a^{5}z^{5}-42a^{5}z^{3}+15a^{5}z-a^{5}z^{-1}+a^{4}z^{10}-4a^{4}z^{8}+10a^{4}z^{4}-6a^{4}z^{2}+2a^{3}z^{9}-13a^{3}z^{7}+27a^{3}z^{5}-21a^{3}z^{3}+5a^{3}z+a^{2}z^{8}-6a^{2}z^{6}+10a^{2}z^{4}-4a^{2}z^{2}$ (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

1

0

-4

3

1

2

-6

2

0

-8

3

2

1

-10

2

0

-12

3

0

-14

1

2

1

-16

2

3

-1

-18

1

2

1

-20

1

-1

-22

1

Integral Khovanov Homology

(db, data source)

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