L11a56

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,8,15,7 X_16,10,17,9 X_20,12,21,11 X_22,18,5,17 X_18,22,19,21 X_8,16,9,15 X_10,20,11,19 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 4, -9, 5, -2, 11, -3, 8, -4, 6, -7, 9, -5, 7, -6}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{4}-2t(2)^{3}+3t(2)^{2}-2t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{23/2}-3q^{21/2}+6q^{19/2}-9q^{17/2}+13q^{15/2}-14q^{13/2}+13q^{11/2}-12q^{9/2}+8q^{7/2}-6q^{5/2}+2q^{3/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-9}+3z^{3}a^{-9}+2za^{-9}+a^{-9}z^{-1}-z^{7}a^{-7}-4z^{5}a^{-7}-5z^{3}a^{-7}-3za^{-7}-a^{-7}z^{-1}-z^{7}a^{-5}-4z^{5}a^{-5}-5z^{3}a^{-5}-4za^{-5}-2a^{-5}z^{-1}+z^{5}a^{-3}+4z^{3}a^{-3}+5za^{-3}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-2z^{9}a^{-5}-6z^{9}a^{-7}-4z^{9}a^{-9}-2z^{8}a^{-4}-3z^{8}a^{-6}-7z^{8}a^{-8}-6z^{8}a^{-10}-z^{7}a^{-3}+3z^{7}a^{-5}+15z^{7}a^{-7}+5z^{7}a^{-9}-6z^{7}a^{-11}+7z^{6}a^{-4}+17z^{6}a^{-6}+25z^{6}a^{-8}+10z^{6}a^{-10}-5z^{6}a^{-12}+5z^{5}a^{-3}+9z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}+8z^{5}a^{-11}-3z^{5}a^{-13}-5z^{4}a^{-4}-15z^{4}a^{-6}-23z^{4}a^{-8}-6z^{4}a^{-10}+6z^{4}a^{-12}-z^{4}a^{-14}-9z^{3}a^{-3}-16z^{3}a^{-5}-2z^{3}a^{-7}-3z^{3}a^{-9}-5z^{3}a^{-11}+3z^{3}a^{-13}-3z^{2}a^{-4}+4z^{2}a^{-6}+10z^{2}a^{-8}-z^{2}a^{-10}-3z^{2}a^{-12}+z^{2}a^{-14}+7za^{-3}+8za^{-5}+za^{-11}+3a^{-4}-3a^{-8}+a^{-12}-2a^{-3}z^{-1}-2a^{-5}z^{-1}+a^{-7}z^{-1}+a^{-9}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

24

1

-1

22

2

20

4

1

-3

18

5

2

3

16

8

4

-4

14

6

5

1

12

7

8

1

10

5

6

-1

8

3

7

4

6

3

5

-2

4

1

5

4

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

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