L11n233

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_9,18,10,19 X_6,13,7,14 X_14,7,15,8 X_8,15,1,16 X_17,22,18,9 X_21,16,22,17 X_19,4,20,5 X_5,20,6,21
Gauss code	{1, -2, 3, 10, -11, -5, 6, -7}, {-4, -1, 2, -3, 5, -6, 7, 9, -8, 4, -10, 11, -9, 8}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-u^{3}v^{5}+u^{2}v^{4}-3u^{2}v^{3}+2u^{2}v^{2}-u^{2}v-uv^{4}+2uv^{3}-3uv^{2}+uv-1}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{13/2}}}-{\frac {1}{q^{15/2}}}+{\frac {3}{q^{17/2}}}-{\frac {4}{q^{19/2}}}+{\frac {5}{q^{21/2}}}-{\frac {5}{q^{23/2}}}+{\frac {4}{q^{25/2}}}-{\frac {3}{q^{27/2}}}+{\frac {1}{q^{29/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-2a^{13}z^{3}-4a^{13}z-a^{13}z^{-1}+a^{11}z^{7}+9a^{11}z^{5}+23a^{11}z^{3}+19a^{11}z+3a^{11}z^{-1}-a^{9}z^{9}-9a^{9}z^{7}-28a^{9}z^{5}-37a^{9}z^{3}-19a^{9}z-2a^{9}z^{-1}$ (db)
Kauffman polynomial	$a^{18}z^{4}-a^{18}z^{2}+3a^{17}z^{5}-5a^{17}z^{3}+a^{17}z+3a^{16}z^{6}-4a^{16}z^{4}+a^{15}z^{7}+3a^{15}z^{5}-7a^{15}z^{3}+3a^{15}z+3a^{14}z^{6}-2a^{14}z^{4}-a^{14}z^{2}+a^{14}+4a^{13}z^{5}-6a^{13}z^{3}+5a^{13}z-a^{13}z^{-1}+a^{12}z^{8}-9a^{12}z^{6}+26a^{12}z^{4}-21a^{12}z^{2}+3a^{12}+a^{11}z^{9}-10a^{11}z^{7}+32a^{11}z^{5}-41a^{11}z^{3}+22a^{11}z-3a^{11}z^{-1}+a^{10}z^{8}-9a^{10}z^{6}+23a^{10}z^{4}-19a^{10}z^{2}+3a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-37a^{9}z^{3}+19a^{9}z-2a^{9}z^{-1}$ (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		\

-11

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-8

1

-10

1

-12

1

-14

2

-16

2

1

-2

-18

3

2

1

-20

2

1

-1

-22

3

0

-24

2

3

1

-26

1

2

-1

-28

2

-30

1

-1

Integral Khovanov Homology

(db, data source)

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