L11a220

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_22,10,7,9 X_10,14,11,13 X_16,6,17,5 X_20,15,21,16 X_18,21,19,22 X_14,19,15,20 X₂₇₃₈ X_4,11,5,12 X_6,18,1,17
Gauss code	{1, -9, 2, -10, 5, -11}, {9, -1, 3, -4, 10, -2, 4, -8, 6, -5, 11, -7, 8, -6, 7, -3}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-4t(1)^{2}t(2)^{3}+8t(1)t(2)^{3}-5t(2)^{3}+7t(1)^{2}t(2)^{2}-13t(1)t(2)^{2}+7t(2)^{2}-5t(1)^{2}t(2)+8t(1)t(2)-4t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {14}{q^{9/2}}}-q^{7/2}+{\frac {18}{q^{7/2}}}+4q^{5/2}-{\frac {23}{q^{5/2}}}-9q^{3/2}+{\frac {22}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {8}{q^{11/2}}}+15{\sqrt {q}}-{\frac {20}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-a^{7}z^{-1}+3z^{3}a^{5}+5za^{5}+3a^{5}z^{-1}-3z^{5}a^{3}-8z^{3}a^{3}-8za^{3}-2a^{3}z^{-1}+z^{7}a+4z^{5}a+8z^{3}a+5za-z^{5}a^{-1}-2z^{3}a^{-1}-2za^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{10}-a^{2}z^{10}-4a^{5}z^{9}-9a^{3}z^{9}-5az^{9}-5a^{6}z^{8}-16a^{4}z^{8}-20a^{2}z^{8}-9z^{8}-3a^{7}z^{7}-2a^{5}z^{7}-a^{3}z^{7}-10az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+9a^{6}z^{6}+39a^{4}z^{6}+43a^{2}z^{6}-4z^{6}a^{-2}+10z^{6}+7a^{7}z^{5}+20a^{5}z^{5}+35a^{3}z^{5}+35az^{5}+12z^{5}a^{-1}-z^{5}a^{-3}+3a^{8}z^{4}-3a^{6}z^{4}-27a^{4}z^{4}-28a^{2}z^{4}+5z^{4}a^{-2}-2z^{4}-6a^{7}z^{3}-20a^{5}z^{3}-33a^{3}z^{3}-28az^{3}-8z^{3}a^{-1}+z^{3}a^{-3}-3a^{8}z^{2}-3a^{6}z^{2}+4a^{4}z^{2}+6a^{2}z^{2}-2z^{2}a^{-2}+3a^{7}z+9a^{5}z+10a^{3}z+7az+3za^{-1}+a^{8}+3a^{6}+3a^{4}-a^{7}z^{-1}-3a^{5}z^{-1}-2a^{3}z^{-1}$ (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

9

3

-6

0

11

6

5

-2

12

10

-2

-4

11

10

1

-6

8

13

5

-8

6

10

-4

-10

2

8

6

-12

1

6

-5

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$