L11a494

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_16,9,17,10 X_8,15,9,16 X_4,17,1,18 X_22,12,19,11 X_10,4,11,3 X_20,5,21,6 X_18,21,5,22 X_12,20,13,19 X_2,14,3,13
Gauss code	{1, -11, 7, -5}, {10, -8, 9, -6}, {8, -1, 2, -4, 3, -7, 6, -10, 11, -2, 4, -3, 5, -9}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)(w-1)^{3}\left(w^{2}+1\right)}{{\sqrt {u}}{\sqrt {v}}w^{5/2}}}$ (db)
Jones polynomial	$q^{3}-4q^{2}+9q-12+19q^{-1}-20q^{-2}+21q^{-3}-17q^{-4}+13q^{-5}-8q^{-6}+3q^{-7}-q^{-8}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{6}z^{4}-3a^{6}z^{2}-a^{6}z^{-2}-3a^{6}+2a^{4}z^{6}+8a^{4}z^{4}+12a^{4}z^{2}+4a^{4}z^{-2}+10a^{4}-a^{2}z^{8}-5a^{2}z^{6}-10a^{2}z^{4}-12a^{2}z^{2}-5a^{2}z^{-2}-11a^{2}+z^{6}+3z^{4}+3z^{2}+2z^{-2}+4$ (db)
Kauffman polynomial	$a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+3a^{8}z^{6}-4a^{8}z^{4}+a^{8}z^{2}+6a^{7}z^{7}-10a^{7}z^{5}+8a^{7}z^{3}-5a^{7}z+a^{7}z^{-1}+7a^{6}z^{8}-10a^{6}z^{6}+8a^{6}z^{4}-9a^{6}z^{2}-a^{6}z^{-2}+5a^{6}+5a^{5}z^{9}+a^{5}z^{7}-20a^{5}z^{5}+32a^{5}z^{3}-22a^{5}z+5a^{5}z^{-1}+2a^{4}z^{10}+11a^{4}z^{8}-38a^{4}z^{6}+52a^{4}z^{4}-38a^{4}z^{2}-4a^{4}z^{-2}+16a^{4}+12a^{3}z^{9}-25a^{3}z^{7}+8a^{3}z^{5}+25a^{3}z^{3}-26a^{3}z+9a^{3}z^{-1}+2a^{2}z^{10}+12a^{2}z^{8}-50a^{2}z^{6}+z^{6}a^{-2}+65a^{2}z^{4}-2z^{4}a^{-2}-40a^{2}z^{2}-5a^{2}z^{-2}+17a^{2}+7az^{9}-16az^{7}+4z^{7}a^{-1}+8az^{5}-9z^{5}a^{-1}+5az^{3}+2z^{3}a^{-1}-10az+5az^{-1}+8z^{8}-24z^{6}+23z^{4}-12z^{2}-2z^{-2}+7$ (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

7

1

5

3

-3

3

6

1

5

1

6

3

-3

-1

13

6

7

-3

11

10

-1

-5

10

9

1

-7

7

11

4

-9

6

10

-4

-11

2

7

5

-13

1

6

-5

-15

2

-17

1

-1

Integral Khovanov Homology

(db, data source)

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