L11a179

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {\left(v^{2}-v+1\right)(uv-u-v+2)(2uv-u-v+1)}{uv^{2}}}$ (db)
Jones polynomial	$24q^{9/2}-25q^{7/2}+20q^{5/2}-16q^{3/2}+{\frac {1}{q^{3/2}}}-q^{19/2}+4q^{17/2}-9q^{15/2}+16q^{13/2}-21q^{11/2}+9{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-7}-2z^{3}a^{-7}-za^{-7}+a^{-7}z^{-1}+z^{7}a^{-5}+3z^{5}a^{-5}+3z^{3}a^{-5}-za^{-5}-3a^{-5}z^{-1}+z^{7}a^{-3}+3z^{5}a^{-3}+4z^{3}a^{-3}+4za^{-3}+2a^{-3}z^{-1}-z^{5}a^{-1}-2z^{3}a^{-1}-za^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-11}-z^{3}a^{-11}+4z^{6}a^{-10}-5z^{4}a^{-10}+z^{2}a^{-10}+8z^{7}a^{-9}-11z^{5}a^{-9}+4z^{3}a^{-9}-za^{-9}+11z^{8}a^{-8}-19z^{6}a^{-8}+14z^{4}a^{-8}-4z^{2}a^{-8}-a^{-8}+9z^{9}a^{-7}-11z^{7}a^{-7}+2z^{5}a^{-7}+4z^{3}a^{-7}-2za^{-7}+a^{-7}z^{-1}+3z^{10}a^{-6}+14z^{8}a^{-6}-44z^{6}a^{-6}+41z^{4}a^{-6}-9z^{2}a^{-6}-3a^{-6}+16z^{9}a^{-5}-32z^{7}a^{-5}+17z^{5}a^{-5}+3z^{3}a^{-5}-5za^{-5}+3a^{-5}z^{-1}+3z^{10}a^{-4}+10z^{8}a^{-4}-36z^{6}a^{-4}+30z^{4}a^{-4}-6z^{2}a^{-4}-3a^{-4}+7z^{9}a^{-3}-9z^{7}a^{-3}-6z^{5}a^{-3}+10z^{3}a^{-3}-6za^{-3}+2a^{-3}z^{-1}+7z^{8}a^{-2}-14z^{6}a^{-2}+6z^{4}a^{-2}-z^{2}a^{-2}+4z^{7}a^{-1}-9z^{5}a^{-1}+6z^{3}a^{-1}-2za^{-1}+z^{6}-2z^{4}+z^{2}$ (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		\

-3

-2

-1

0

1

2

3

4

5

6

7

8

χ

20

1

18

3

-3

16

6

1

5

14

10

3

-7

12

11

6

5

10

13

10

-3

8

12

11

1

6

9

14

5

4

7

11

-4

2

3

10

7

0

1

6

-5

-2

3

-4

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=3$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=5$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=7$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=8$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$