L11a256

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{4}-2u^{3}v^{3}+2u^{3}v^{2}-u^{3}v+u^{2}v^{5}-4u^{2}v^{4}+7u^{2}v^{3}-8u^{2}v^{2}+4u^{2}v-u^{2}-uv^{5}+4uv^{4}-8uv^{3}+7uv^{2}-4uv+u-v^{4}+2v^{3}-2v^{2}+v}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}-q^{7/2}+{\frac {16}{q^{7/2}}}+4q^{5/2}-{\frac {20}{q^{5/2}}}-9q^{3/2}+{\frac {20}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {6}{q^{11/2}}}+14{\sqrt {q}}-{\frac {18}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+az^{7}-a^{5}z^{5}+4a^{3}z^{5}+3az^{5}-z^{5}a^{-1}-3a^{5}z^{3}+6a^{3}z^{3}+2az^{3}-2z^{3}a^{-1}-2a^{5}z+2a^{3}z-az-za^{-1}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-2a^{4}z^{10}-2a^{2}z^{10}-4a^{5}z^{9}-10a^{3}z^{9}-6az^{9}-4a^{6}z^{8}-3a^{4}z^{8}-8a^{2}z^{8}-9z^{8}-3a^{7}z^{7}+6a^{5}z^{7}+21a^{3}z^{7}+4az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+8a^{6}z^{6}+9a^{4}z^{6}+18a^{2}z^{6}-4z^{6}a^{-2}+14z^{6}+9a^{7}z^{5}-a^{5}z^{5}-22a^{3}z^{5}+2az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}-2a^{6}z^{4}-4a^{4}z^{4}-11a^{2}z^{4}+5z^{4}a^{-2}-7z^{4}-8a^{7}z^{3}-3a^{5}z^{3}+13a^{3}z^{3}+az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}-2a^{8}z^{2}-2a^{6}z^{2}-a^{4}z^{2}+2a^{2}z^{2}-z^{2}a^{-2}+2z^{2}+2a^{7}z+4a^{5}z-a^{3}z-2az+za^{-1}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

8

3

-5

0

10

6

4

-2

11

9

-2

-4

9

0

-6

7

11

4

-8

5

9

-4

-10

2

8

6

-12

1

4

-3

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$