L11a368

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Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X_14,3,15,4 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_4,17,5,18 X_16,5,17,6 X_6,15,7,16
Gauss code	{1, -2, 3, -9, 10, -11, 4, -5, 6, -8}, {5, -1, 2, -3, 11, -10, 9, -4, 7, -6, 8, -7}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{4}v^{3}-u^{4}v^{2}+u^{3}v^{4}-3u^{3}v^{3}+4u^{3}v^{2}-u^{3}v-u^{2}v^{4}+4u^{2}v^{3}-5u^{2}v^{2}+4u^{2}v-u^{2}-uv^{3}+4uv^{2}-3uv+u-v^{2}+v}{u^{2}v^{2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {12}{q^{7/2}}}+{\frac {11}{q^{9/2}}}-{\frac {10}{q^{11/2}}}+{\frac {7}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+4za^{7}+a^{7}z^{-1}-z^{7}a^{5}-5z^{5}a^{5}-9z^{3}a^{5}-7za^{5}-a^{5}z^{-1}-z^{7}a^{3}-4z^{5}a^{3}-4z^{3}a^{3}-za^{3}+z^{5}a+3z^{3}a+za$ (db)
Kauffman polynomial	$a^{11}z^{5}-3a^{11}z^{3}+2a^{11}z+2a^{10}z^{6}-5a^{10}z^{4}+3a^{10}z^{2}+2a^{9}z^{7}-2a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+2a^{8}z^{8}-2a^{8}z^{6}+a^{8}z^{4}-a^{8}z^{2}+2a^{7}z^{9}-5a^{7}z^{7}+13a^{7}z^{5}-15a^{7}z^{3}+7a^{7}z-a^{7}z^{-1}+a^{6}z^{10}-3a^{6}z^{6}+10a^{6}z^{4}-7a^{6}z^{2}+a^{6}+5a^{5}z^{9}-18a^{5}z^{7}+31a^{5}z^{5}-24a^{5}z^{3}+9a^{5}z-a^{5}z^{-1}+a^{4}z^{10}+2a^{4}z^{8}-13a^{4}z^{6}+17a^{4}z^{4}-6a^{4}z^{2}+3a^{3}z^{9}-8a^{3}z^{7}+5a^{3}z^{5}-a^{3}z^{3}+4a^{2}z^{8}-13a^{2}z^{6}+10a^{2}z^{4}-2a^{2}z^{2}+3az^{7}-10az^{5}+7az^{3}-az+z^{6}-3z^{4}+z^{2}$ (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

4

1

-1

2

0

3

1

-2

5

2

3

-4

6

4

-2

-6

6

4

2

-8

5

6

1

-10

5

6

-1

-12

2

5

3

-14

2

5

-3

-16

1

3

2

-18

1

-1

-20

1

Integral Khovanov Homology

(db, data source)

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