L11a204

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{4}-4u^{2}v^{3}+4u^{2}v^{2}-u^{2}v-uv^{4}+5uv^{3}-7uv^{2}+5uv-u-v^{3}+4v^{2}-4v+2}{uv^{2}}}$ (db)
Jones polynomial	$q^{3/2}-2{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {10}{q^{5/2}}}-{\frac {13}{q^{7/2}}}+{\frac {13}{q^{9/2}}}-{\frac {12}{q^{11/2}}}+{\frac {9}{q^{13/2}}}-{\frac {6}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+3z^{3}a^{7}+2za^{7}-z^{7}a^{5}-4z^{5}a^{5}-4z^{3}a^{5}+za^{5}+2a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-10z^{3}a^{3}-10za^{3}-3a^{3}z^{-1}+z^{5}a+4z^{3}a+4za+az^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+2z^{3}a^{11}-za^{11}-3z^{6}a^{10}+6z^{4}a^{10}-2z^{2}a^{10}-4z^{7}a^{9}+6z^{5}a^{9}-4z^{8}a^{8}+6z^{6}a^{8}-3z^{4}a^{8}+2z^{2}a^{8}-3z^{9}a^{7}+5z^{7}a^{7}-7z^{5}a^{7}+6z^{3}a^{7}-3za^{7}-z^{10}a^{6}-3z^{8}a^{6}+11z^{6}a^{6}-16z^{4}a^{6}+6z^{2}a^{6}-5z^{9}a^{5}+14z^{7}a^{5}-17z^{5}a^{5}+3z^{3}a^{5}+4za^{5}-2a^{5}z^{-1}-z^{10}a^{4}-z^{8}a^{4}+6z^{6}a^{4}-5z^{4}a^{4}-3z^{2}a^{4}+3a^{4}-2z^{9}a^{3}+3z^{7}a^{3}+4z^{5}a^{3}-12z^{3}a^{3}+12za^{3}-3a^{3}z^{-1}-2z^{8}a^{2}+3z^{6}a^{2}+6z^{4}a^{2}-9z^{2}a^{2}+3a^{2}-2z^{7}a+7z^{5}a-7z^{3}a+4za-az^{-1}-z^{6}+4z^{4}-4z^{2}+1$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

1

0

3

1

-2

5

1

4

-4

6

4

-2

-6

7

4

3

-8

6

0

-10

6

7

-1

-12

3

6

3

-14

3

6

-3

-16

1

4

3

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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