L11n190

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_22,10,7,9 X_10,14,11,13 X_5,16,6,17 X_20,15,21,16 X_18,21,19,22 X_14,19,15,20 X₂₇₃₈ X_4,11,5,12 X_17,6,18,1
Gauss code	{1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, -8, 6, 5, -11, -7, 8, -6, 7, -3}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {-u^{2}v^{3}+4u^{2}v^{2}-3u^{2}v+u^{2}+4uv^{3}-7uv^{2}+4uv+v^{4}-3v^{3}+4v^{2}-v}{uv^{2}}}$ (db)
Jones polynomial	${\frac {10}{q^{9/2}}}-{\frac {8}{q^{7/2}}}+{\frac {5}{q^{5/2}}}-{\frac {2}{q^{3/2}}}+{\frac {1}{q^{21/2}}}-{\frac {3}{q^{19/2}}}+{\frac {6}{q^{17/2}}}-{\frac {9}{q^{15/2}}}+{\frac {10}{q^{13/2}}}-{\frac {12}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{11}z^{-1}+4a^{9}z+3a^{9}z^{-1}-4a^{7}z^{3}-6a^{7}z-2a^{7}z^{-1}+a^{5}z^{5}+a^{5}z^{3}+a^{5}z-2a^{3}z^{3}-2a^{3}z$ (db)
Kauffman polynomial	$-z^{6}a^{12}+3z^{4}a^{12}-3z^{2}a^{12}+a^{12}-3z^{7}a^{11}+9z^{5}a^{11}-8z^{3}a^{11}+3za^{11}-a^{11}z^{-1}-3z^{8}a^{10}+4z^{6}a^{10}+8z^{4}a^{10}-11z^{2}a^{10}+3a^{10}-z^{9}a^{9}-7z^{7}a^{9}+26z^{5}a^{9}-24z^{3}a^{9}+12za^{9}-3a^{9}z^{-1}-6z^{8}a^{8}+8z^{6}a^{8}+7z^{4}a^{8}-9z^{2}a^{8}+3a^{8}-z^{9}a^{7}-7z^{7}a^{7}+19z^{5}a^{7}-16z^{3}a^{7}+7za^{7}-2a^{7}z^{-1}-3z^{8}a^{6}+2z^{6}a^{6}-2z^{4}a^{6}+2z^{2}a^{6}-3z^{7}a^{5}+2z^{5}a^{5}-3z^{3}a^{5}-z^{6}a^{4}-4z^{4}a^{4}+3z^{2}a^{4}-3z^{3}a^{3}+2za^{3}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

2

-4

4

1

-3

-6

4

1

3

-8

6

4

-2

-10

6

4

2

-12

4

6

2

-14

5

6

-1

-16

2

5

3

-18

1

4

-3

-20

2

-22

1

-1

Integral Khovanov Homology

(db, data source)

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