L11a343

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Planar diagram presentation	X_12,1,13,2 X_16,8,17,7 X_10,5,1,6 X₆₃₇₄ X_4,9,5,10 X_18,16,19,15 X_22,19,11,20 X_20,13,21,14 X_14,21,15,22 X_2,11,3,12 X_8,18,9,17
Gauss code	{1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, 8, -9, 6, -2, 11, -6, 7, -8, 9, -7}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}+t(1)^{2}-2t(2)^{2}t(1)+3t(2)t(1)-2t(1)+t(2)^{2}-2t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{3/2}-4{\sqrt {q}}+{\frac {8}{\sqrt {q}}}-{\frac {14}{q^{3/2}}}+{\frac {16}{q^{5/2}}}-{\frac {20}{q^{7/2}}}+{\frac {19}{q^{9/2}}}-{\frac {16}{q^{11/2}}}+{\frac {12}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-3a^{7}z^{3}-7a^{7}z-5a^{7}z^{-1}+3a^{5}z^{5}+10a^{5}z^{3}+14a^{5}z+8a^{5}z^{-1}-a^{3}z^{7}-4a^{3}z^{5}-8a^{3}z^{3}-9a^{3}z-4a^{3}z^{-1}+az^{5}+2az^{3}+az$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-6a^{10}z^{4}+5a^{10}z^{2}-2a^{10}+4a^{9}z^{7}-3a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+a^{9}z^{-1}+4a^{8}z^{8}+3a^{8}z^{6}-18a^{8}z^{4}+22a^{8}z^{2}-9a^{8}+3a^{7}z^{9}+6a^{7}z^{7}-17a^{7}z^{5}+16a^{7}z^{3}-9a^{7}z+5a^{7}z^{-1}+a^{6}z^{10}+10a^{6}z^{8}-13a^{6}z^{6}-12a^{6}z^{4}+29a^{6}z^{2}-14a^{6}+7a^{5}z^{9}-27a^{5}z^{5}+33a^{5}z^{3}-21a^{5}z+8a^{5}z^{-1}+a^{4}z^{10}+12a^{4}z^{8}-26a^{4}z^{6}+5a^{4}z^{4}+13a^{4}z^{2}-8a^{4}+4a^{3}z^{9}+2a^{3}z^{7}-24a^{3}z^{5}+25a^{3}z^{3}-14a^{3}z+4a^{3}z^{-1}+6a^{2}z^{8}-12a^{2}z^{6}+3a^{2}z^{4}+2a^{2}z^{2}+4az^{7}-10az^{5}+8az^{3}-2az+z^{6}-2z^{4}+z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

3

0

5

1

-4

-2

9

3

6

-4

9

7

-2

-6

11

7

4

-8

8

9

1

-10

8

11

-3

-12

4

8

4

-14

2

8

-6

-16

1

4

3

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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