L11n168

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_18,11,19,12 X_19,7,20,22 X_15,21,16,20 X_21,17,22,16 X_12,17,13,18 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -9, 11, -4, -7, 8, 9, -5, -6, 7, -8, 6}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-t(1)^{2}t(2)^{3}+3t(1)t(2)^{3}-t(2)^{3}+2t(1)^{2}t(2)^{2}-3t(1)t(2)^{2}+2t(2)^{2}-t(1)^{2}t(2)+3t(1)t(2)-t(2)-t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {7}{q^{9/2}}}-{\frac {7}{q^{7/2}}}+{\frac {4}{q^{5/2}}}-{\frac {3}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {2}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {7}{q^{11/2}}}+{\frac {1}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+5za^{7}+a^{7}z^{-1}-z^{7}a^{5}-6z^{5}a^{5}-13z^{3}a^{5}-10za^{5}-a^{5}z^{-1}+z^{5}a^{3}+3z^{3}a^{3}+2za^{3}$ (db)
Kauffman polynomial	$a^{11}z^{5}-3a^{11}z^{3}+2a^{11}z+2a^{10}z^{6}-5a^{10}z^{4}+2a^{10}z^{2}+2a^{9}z^{7}-3a^{9}z^{5}-a^{9}z^{3}+2a^{8}z^{8}-5a^{8}z^{6}+7a^{8}z^{4}-5a^{8}z^{2}+a^{7}z^{9}-2a^{7}z^{7}+5a^{7}z^{5}-6a^{7}z^{3}+5a^{7}z-a^{7}z^{-1}+3a^{6}z^{8}-10a^{6}z^{6}+18a^{6}z^{4}-10a^{6}z^{2}+a^{6}+a^{5}z^{9}-4a^{5}z^{7}+12a^{5}z^{5}-14a^{5}z^{3}+9a^{5}z-a^{5}z^{-1}+a^{4}z^{8}-3a^{4}z^{6}+7a^{4}z^{4}-5a^{4}z^{2}+3a^{3}z^{5}-6a^{3}z^{3}+2a^{3}z+a^{2}z^{4}-2a^{2}z^{2}$ (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

χ

0

1

-1

-2

2

-4

3

2

-1

-6

4

1

3

-8

3

0

-10

4

0

-12

2

3

1

-14

2

4

-2

-16

1

3

2

-18

1

-1

-20

1

Integral Khovanov Homology

(db, data source)

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