L11a169

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{4}-6u^{2}v^{3}+6u^{2}v^{2}-2u^{2}v-2uv^{4}+9uv^{3}-15uv^{2}+9uv-2u-2v^{3}+6v^{2}-6v+2}{uv^{2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+14{\sqrt {q}}-{\frac {20}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {22}{q^{5/2}}}+{\frac {19}{q^{7/2}}}-{\frac {14}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-2a^{5}z^{3}+a^{5}z^{-1}+a^{3}z^{7}+3a^{3}z^{5}+2a^{3}z^{3}-2a^{3}z-2a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+3az^{3}-2z^{3}a^{-1}+2az-za^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3a^{4}z^{10}-3a^{2}z^{10}-7a^{5}z^{9}-15a^{3}z^{9}-8az^{9}-7a^{6}z^{8}-6a^{4}z^{8}-9a^{2}z^{8}-10z^{8}-4a^{7}z^{7}+15a^{5}z^{7}+38a^{3}z^{7}+11az^{7}-8z^{7}a^{-1}-a^{8}z^{6}+18a^{6}z^{6}+31a^{4}z^{6}+34a^{2}z^{6}-4z^{6}a^{-2}+18z^{6}+10a^{7}z^{5}-8a^{5}z^{5}-35a^{3}z^{5}-3az^{5}+13z^{5}a^{-1}-z^{5}a^{-3}+2a^{8}z^{4}-13a^{6}z^{4}-35a^{4}z^{4}-38a^{2}z^{4}+5z^{4}a^{-2}-13z^{4}-5a^{7}z^{3}+6a^{3}z^{3}-6az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}+4a^{6}z^{2}+13a^{4}z^{2}+14a^{2}z^{2}+5z^{2}+2a^{5}z+6a^{3}z+6az+2za^{-1}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

8

3

-5

0

12

6

-2

11

9

-2

-4

11

0

-6

9

12

3

-8

5

10

-5

-10

3

9

6

-12

1

5

-4

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$