L11a147

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_18,12,19,11 X_12,6,13,5 X_4,19,5,20 X_16,7,17,8 X_20,13,21,14 X_22,15,7,16 X_14,21,15,22 X_6,18,1,17
Gauss code	{1, -2, 3, -6, 5, -11}, {7, -1, 2, -3, 4, -5, 8, -10, 9, -7, 11, -4, 6, -8, 10, -9}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{6}-2u^{2}v^{5}+2u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}-u^{2}v-uv^{6}+3uv^{5}-3uv^{4}+3uv^{3}-3uv^{2}+3uv-u-v^{5}+2v^{4}-2v^{3}+2v^{2}-2v+1}{uv^{3}}}$ (db)
Jones polynomial	$-{\frac {11}{q^{9/2}}}+{\frac {9}{q^{7/2}}}-{\frac {8}{q^{5/2}}}+{\frac {4}{q^{3/2}}}-{\frac {1}{q^{21/2}}}+{\frac {3}{q^{19/2}}}-{\frac {5}{q^{17/2}}}+{\frac {8}{q^{15/2}}}-{\frac {10}{q^{13/2}}}+{\frac {11}{q^{11/2}}}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$-a^{5}z^{9}+a^{7}z^{7}-7a^{5}z^{7}+a^{3}z^{7}+5a^{7}z^{5}-17a^{5}z^{5}+5a^{3}z^{5}+7a^{7}z^{3}-16a^{5}z^{3}+6a^{3}z^{3}+2a^{7}z-2a^{5}z-a^{3}z-a^{7}z^{-1}+3a^{5}z^{-1}-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{13}-3z^{4}a^{12}+z^{2}a^{12}-5z^{5}a^{11}+3z^{3}a^{11}-7z^{6}a^{10}+9z^{4}a^{10}-2z^{2}a^{10}-8z^{7}a^{9}+16z^{5}a^{9}-7z^{3}a^{9}+za^{9}-7z^{8}a^{8}+16z^{6}a^{8}-4z^{4}a^{8}-3z^{2}a^{8}+a^{8}-5z^{9}a^{7}+13z^{7}a^{7}-4z^{5}a^{7}-z^{3}a^{7}-a^{7}z^{-1}-2z^{10}a^{6}+z^{8}a^{6}+17z^{6}a^{6}-20z^{4}a^{6}+z^{2}a^{6}+3a^{6}-8z^{9}a^{5}+38z^{7}a^{5}-56z^{5}a^{5}+29z^{3}a^{5}-za^{5}-3a^{5}z^{-1}-2z^{10}a^{4}+7z^{8}a^{4}-z^{6}a^{4}-11z^{4}a^{4}+3z^{2}a^{4}+3a^{4}-3z^{9}a^{3}+17z^{7}a^{3}-31z^{5}a^{3}+19z^{3}a^{3}-2a^{3}z^{-1}-z^{8}a^{2}+5z^{6}a^{2}-7z^{4}a^{2}+2z^{2}a^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

2

1

-1

0

2

-2

2

1

-1

-4

6

2

4

-6

4

3

-1

-8

7

5

2

-10

5

0

-12

5

6

-1

-14

3

5

2

-16

2

5

-3

-18

1

3

2

-20

2

-2

-22

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-6$	$i=-4$
$r=-8$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$