L11a184

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {\left(t(1)t(2)^{2}-t(2)^{2}-2t(1)t(2)+2t(2)+t(1)-2\right)\left(2t(1)t(2)^{2}-t(2)^{2}-2t(1)t(2)+2t(2)+t(1)-1\right)}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+16{\sqrt {q}}-{\frac {23}{\sqrt {q}}}+{\frac {25}{q^{3/2}}}-{\frac {27}{q^{5/2}}}+{\frac {23}{q^{7/2}}}-{\frac {17}{q^{9/2}}}+{\frac {11}{q^{11/2}}}-{\frac {5}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-a^{5}z^{3}+a^{5}z+a^{3}z^{7}+2a^{3}z^{5}-a^{3}z^{3}-3a^{3}z+a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+4az^{3}-2z^{3}a^{-1}+2az-za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-a^{8}z^{4}+5a^{7}z^{7}-9a^{7}z^{5}+3a^{7}z^{3}+10a^{6}z^{8}-22a^{6}z^{6}+12a^{6}z^{4}+9a^{5}z^{9}-12a^{5}z^{7}-5a^{5}z^{5}+7a^{5}z^{3}-2a^{5}z+3a^{4}z^{10}+16a^{4}z^{8}-47a^{4}z^{6}+29a^{4}z^{4}-2a^{4}z^{2}+17a^{3}z^{9}-26a^{3}z^{7}+z^{5}a^{-3}+12a^{3}z^{3}-z^{3}a^{-3}-5a^{3}z-a^{3}z^{-1}+3a^{2}z^{10}+16a^{2}z^{8}-39a^{2}z^{6}+4z^{6}a^{-2}+23a^{2}z^{4}-5z^{4}a^{-2}-2a^{2}z^{2}+2z^{2}a^{-2}+a^{2}+8az^{9}-az^{7}+8z^{7}a^{-1}-16az^{5}-11z^{5}a^{-1}+16az^{3}+7z^{3}a^{-1}-5az-2za^{-1}-az^{-1}+10z^{8}-11z^{6}+2z^{4}+2z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

10

3

-7

0

13

6

7

-2

13

11

-2

-4

14

12

2

-6

10

14

4

-8

7

13

-6

-10

4

10

6

-12

1

7

-6

-14

4

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=-2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$