L11a193

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2t(1)^{2}t(2)^{4}-3t(1)t(2)^{4}+t(2)^{4}-4t(1)^{2}t(2)^{3}+6t(1)t(2)^{3}-3t(2)^{3}+4t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+4t(2)^{2}-3t(1)^{2}t(2)+6t(1)t(2)-4t(2)+t(1)^{2}-3t(1)+2}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-14q^{9/2}+9q^{7/2}-6q^{5/2}+2q^{3/2}+q^{23/2}-4q^{21/2}+8q^{19/2}-12q^{17/2}+16q^{15/2}-17q^{13/2}+16q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$z^{5}a^{-9}+2z^{3}a^{-9}-z^{7}a^{-7}-3z^{5}a^{-7}-z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}-z^{7}a^{-5}-4z^{5}a^{-5}-6z^{3}a^{-5}-6za^{-5}-3a^{-5}z^{-1}+z^{5}a^{-3}+4z^{3}a^{-3}+5za^{-3}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-6}-z^{10}a^{-8}-2z^{9}a^{-5}-6z^{9}a^{-7}-4z^{9}a^{-9}-2z^{8}a^{-4}-4z^{8}a^{-6}-10z^{8}a^{-8}-8z^{8}a^{-10}-z^{7}a^{-3}+2z^{7}a^{-5}+7z^{7}a^{-7}-6z^{7}a^{-9}-10z^{7}a^{-11}+7z^{6}a^{-4}+16z^{6}a^{-6}+22z^{6}a^{-8}+5z^{6}a^{-10}-8z^{6}a^{-12}+5z^{5}a^{-3}+12z^{5}a^{-5}+15z^{5}a^{-7}+24z^{5}a^{-9}+12z^{5}a^{-11}-4z^{5}a^{-13}-6z^{4}a^{-4}-8z^{4}a^{-6}-3z^{4}a^{-8}+8z^{4}a^{-10}+8z^{4}a^{-12}-z^{4}a^{-14}-9z^{3}a^{-3}-23z^{3}a^{-5}-21z^{3}a^{-7}-14z^{3}a^{-9}-5z^{3}a^{-11}+2z^{3}a^{-13}-2z^{2}a^{-4}-6z^{2}a^{-6}-6z^{2}a^{-8}-5z^{2}a^{-10}-3z^{2}a^{-12}+7za^{-3}+13za^{-5}+8za^{-7}+3za^{-9}+za^{-11}+3a^{-4}+3a^{-6}+a^{-8}-2a^{-3}z^{-1}-3a^{-5}z^{-1}-a^{-7}z^{-1}$ (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

3

20

5

1

-4

18

7

3

4

16

9

5

-4

14

8

7

1

12

8

9

1

10

6

8

-2

8

4

9

5

6

2

5

-3

4

1

5

4

2

1

-1

0

1

Integral Khovanov Homology

(db, data source)

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