L11a446

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Planar diagram presentation	X₆₁₇₂ X_14,4,15,3 X_18,9,19,10 X_16,8,17,7 X_22,17,13,18 X_20,11,21,12 X_8,14,9,13 X_10,19,11,20 X_12,21,5,22 X₂₅₃₆ X_4,16,1,15
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -7, 3, -8, 6, -9}, {7, -2, 11, -4, 5, -3, 8, -6, 9, -5}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{3}w^{3}-uv^{3}w^{2}-2uv^{2}w^{3}+4uv^{2}w^{2}-2uv^{2}w+uvw^{3}-3uvw^{2}+4uvw-uv+uw^{2}-2uw+u-v^{3}w^{3}+2v^{3}w^{2}-v^{3}w+v^{2}w^{3}-4v^{2}w^{2}+3v^{2}w-v^{2}+2vw^{2}-4vw+2v+w-1}{{\sqrt {u}}v^{3/2}w^{3/2}}}$ (db)
Jones polynomial	$q^{3}-3q^{2}+6q-8+13q^{-1}-14q^{-2}+15q^{-3}-12q^{-4}+10q^{-5}-6q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+9a^{4}z^{4}-14a^{2}z^{4}+4z^{4}-3a^{6}z^{2}+13a^{4}z^{2}-16a^{2}z^{2}+5z^{2}-2a^{6}+7a^{4}-9a^{2}+4+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+3a^{8}z^{6}-6a^{8}z^{4}+3a^{8}z^{2}-a^{8}+4a^{7}z^{7}-5a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+4a^{6}z^{8}-4a^{6}z^{6}-2a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{9}-2a^{5}z^{7}-3a^{5}z^{5}+6a^{5}z^{3}-3a^{5}z+a^{4}z^{10}+7a^{4}z^{8}-29a^{4}z^{6}+47a^{4}z^{4}-31a^{4}z^{2}-a^{4}z^{-2}+9a^{4}+7a^{3}z^{9}-20a^{3}z^{7}+19a^{3}z^{5}+4a^{3}z^{3}-8a^{3}z+2a^{3}z^{-1}+a^{2}z^{10}+8a^{2}z^{8}-41a^{2}z^{6}+z^{6}a^{-2}+67a^{2}z^{4}-3z^{4}a^{-2}-44a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+13a^{2}+4az^{9}-11az^{7}+3z^{7}a^{-1}+7az^{5}-9z^{5}a^{-1}+2az^{3}+4z^{3}a^{-1}-5az+2az^{-1}+5z^{8}-18z^{6}+21z^{4}-14z^{2}-z^{-2}+6$ (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

7

1

5

2

-2

3

4

1

3

1

4

2

-2

-1

9

4

5

-3

8

7

-1

-5

7

6

1

-7

5

8

3

-9

5

7

-2

-11

2

6

4

-13

1

4

-3

-15

2

-17

1

-1

Integral Khovanov Homology

(db, data source)

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