L11a199

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2u^{2}v^{4}-6u^{2}v^{3}+4u^{2}v^{2}-u^{2}v-uv^{4}+5uv^{3}-9uv^{2}+5uv-u-v^{3}+4v^{2}-6v+2}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {12}{q^{9/2}}}+{\frac {8}{q^{7/2}}}-{\frac {5}{q^{5/2}}}+{\frac {2}{q^{3/2}}}+{\frac {1}{q^{23/2}}}-{\frac {4}{q^{21/2}}}+{\frac {7}{q^{19/2}}}-{\frac {11}{q^{17/2}}}+{\frac {14}{q^{15/2}}}-{\frac {15}{q^{13/2}}}+{\frac {14}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-z^{5}a^{9}-2z^{3}a^{9}+za^{9}+a^{9}z^{-1}+z^{7}a^{7}+3z^{5}a^{7}-4za^{7}-2a^{7}z^{-1}+z^{7}a^{5}+4z^{5}a^{5}+5z^{3}a^{5}+4za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{4}a^{14}-4z^{5}a^{13}+3z^{3}a^{13}-7z^{6}a^{12}+7z^{4}a^{12}-z^{2}a^{12}-8z^{7}a^{11}+8z^{5}a^{11}+z^{3}a^{11}-2za^{11}-7z^{8}a^{10}+8z^{6}a^{10}-2z^{2}a^{10}-4z^{9}a^{9}+z^{7}a^{9}+9z^{5}a^{9}-7z^{3}a^{9}+3za^{9}-a^{9}z^{-1}-z^{10}a^{8}-8z^{8}a^{8}+26z^{6}a^{8}-22z^{4}a^{8}+6z^{2}a^{8}-6z^{9}a^{7}+14z^{7}a^{7}-2z^{5}a^{7}-12z^{3}a^{7}+9za^{7}-2a^{7}z^{-1}-z^{10}a^{6}-3z^{8}a^{6}+19z^{6}a^{6}-23z^{4}a^{6}+10z^{2}a^{6}-a^{6}-2z^{9}a^{5}+4z^{7}a^{5}+6z^{5}a^{5}-15z^{3}a^{5}+9za^{5}-2a^{5}z^{-1}-2z^{8}a^{4}+8z^{6}a^{4}-9z^{4}a^{4}+3z^{2}a^{4}-z^{7}a^{3}+5z^{5}a^{3}-8z^{3}a^{3}+5za^{3}-a^{3}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

1

-1

-4

4

1

3

-6

5

2

-3

-8

7

3

4

-10

7

5

-2

-12

8

7

1

-14

6

7

1

-16

5

8

-3

-18

3

7

4

-20

1

4

-3

-22

3

-24

1

-1

Integral Khovanov Homology

(db, data source)

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