L11a166

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X₆₇₁₈ X_16,13,17,14 X_14,6,15,5 X_4,16,5,15 X_20,11,21,12 X_22,18,7,17 X_18,22,19,21 X_12,19,13,20
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 8, -11, 5, -6, 7, -5, 9, -10, 11, -8, 10, -9}

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Polynomial invariants

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

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

1

-1

2

4

1

3

0

4

1

-3

-2

7

4

3

-4

7

5

-2

-6

7

6

1

-8

6

8

2

-10

4

6

-2

-12

2

6

4

-14

1

4

-3

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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