L11a144

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,8,15,7 X_20,14,21,13 X_6,19,1,20 X_18,11,19,12 X_12,6,13,5 X_22,16,7,15 X_4,18,5,17 X_16,22,17,21
Gauss code	{1, -2, 3, -10, 8, -6}, {4, -1, 2, -3, 7, -8, 5, -4, 9, -11, 10, -7, 6, -5, 11, -9}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{4}-5u^{2}v^{3}+5u^{2}v^{2}-2u^{2}v-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u-2v^{3}+5v^{2}-5v+2}{uv^{2}}}$ (db)
Jones polynomial	$3q^{9/2}-{\frac {4}{q^{9/2}}}-6q^{7/2}+{\frac {8}{q^{7/2}}}+11q^{5/2}-{\frac {13}{q^{5/2}}}-15q^{3/2}+{\frac {16}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+17{\sqrt {q}}-{\frac {19}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+4z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+az^{3}+6z^{3}a^{-1}-3z^{3}a^{-3}-3az+4za^{-1}-2za^{-3}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-6az^{9}-10z^{9}a^{-1}-4z^{9}a^{-3}-10a^{2}z^{8}-z^{8}a^{-2}-3z^{8}a^{-4}-8z^{8}-11a^{3}z^{7}+2az^{7}+28z^{7}a^{-1}+14z^{7}a^{-3}-z^{7}a^{-5}-8a^{4}z^{6}+13a^{2}z^{6}+18z^{6}a^{-2}+12z^{6}a^{-4}+27z^{6}-4a^{5}z^{5}+15a^{3}z^{5}+14az^{5}-25z^{5}a^{-1}-16z^{5}a^{-3}+4z^{5}a^{-5}-a^{6}z^{4}+7a^{4}z^{4}-2a^{2}z^{4}-22z^{4}a^{-2}-15z^{4}a^{-4}-17z^{4}+2a^{5}z^{3}-7a^{3}z^{3}-8az^{3}+14z^{3}a^{-1}+9z^{3}a^{-3}-4z^{3}a^{-5}-a^{4}z^{2}-a^{2}z^{2}+9z^{2}a^{-2}+6z^{2}a^{-4}+3z^{2}-2az-5za^{-1}-3za^{-3}-a^{2}+a^{3}z^{-1}+az^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

2

-2

8

4

1

3

6

7

2

-5

4

8

4

2

9

7

-2

0

10

8

2

-2

7

10

3

-4

6

9

-3

-6

3

8

5

-8

1

5

-4

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

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