L11a224

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Planar diagram presentation	X₈₁₉₂ X_16,6,17,5 X_18,10,19,9 X_10,20,11,19 X_2,11,3,12 X_12,3,13,4 X₄₇₅₈ X_20,14,21,13 X_22,16,7,15 X_6,18,1,17 X_14,22,15,21
Gauss code	{1, -5, 6, -7, 2, -10}, {7, -1, 3, -4, 5, -6, 8, -11, 9, -2, 10, -3, 4, -8, 11, -9}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{6}-2u^{2}v^{5}+3u^{2}v^{4}-3u^{2}v^{3}+2u^{2}v^{2}-u^{2}v-uv^{6}+4uv^{5}-6uv^{4}+7uv^{3}-6uv^{2}+4uv-u-v^{5}+2v^{4}-3v^{3}+3v^{2}-2v+1}{uv^{3}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+7q^{13/2}-11q^{11/2}+15q^{9/2}-17q^{7/2}+16q^{5/2}-15q^{3/2}+10{\sqrt {q}}-{\frac {7}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-5}+5z^{5}a^{-5}+9z^{3}a^{-5}+7za^{-5}+2a^{-5}z^{-1}-z^{9}a^{-3}-7z^{7}a^{-3}-19z^{5}a^{-3}-25z^{3}a^{-3}-16za^{-3}-5a^{-3}z^{-1}+z^{7}a^{-1}+5z^{5}a^{-1}+9z^{3}a^{-1}+8za^{-1}+3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}-z^{2}a^{-10}+3z^{5}a^{-9}-2z^{3}a^{-9}+6z^{6}a^{-8}-7z^{4}a^{-8}+5z^{2}a^{-8}-a^{-8}+8z^{7}a^{-7}-11z^{5}a^{-7}+7z^{3}a^{-7}+8z^{8}a^{-6}-12z^{6}a^{-6}+6z^{4}a^{-6}+6z^{9}a^{-5}-10z^{7}a^{-5}+10z^{5}a^{-5}-15z^{3}a^{-5}+9za^{-5}-2a^{-5}z^{-1}+2z^{10}a^{-4}+6z^{8}a^{-4}-26z^{6}a^{-4}+27z^{4}a^{-4}-18z^{2}a^{-4}+5a^{-4}+10z^{9}a^{-3}-32z^{7}a^{-3}+40z^{5}a^{-3}-37z^{3}a^{-3}+20za^{-3}-5a^{-3}z^{-1}+2z^{10}a^{-2}+z^{8}a^{-2}-19z^{6}a^{-2}+24z^{4}a^{-2}-14z^{2}a^{-2}+5a^{-2}+4z^{9}a^{-1}+az^{7}-13z^{7}a^{-1}-4az^{5}+12z^{5}a^{-1}+5az^{3}-8z^{3}a^{-1}-2az+9za^{-1}-3a^{-1}z^{-1}+3z^{8}-11z^{6}+11z^{4}-2z^{2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

2

14

5

1

-4

12

6

2

4

10

9

5

-4

8

6

2

6

8

9

1

4

7

8

-1

2

4

9

5

0

3

6

-3

-2

1

5

4

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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