L11n229

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_12,3,13,4 X_20,5,21,6 X_7,14,8,15 X_13,16,14,17 X_15,8,16,1 X_22,17,9,18 X_18,21,19,22 X_6,9,7,10 X_4,19,5,20
Gauss code	{1, -2, 3, -11, 4, -10, -5, 7}, {10, -1, 2, -3, -6, 5, -7, 6, 8, -9, 11, -4, 9, -8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(2uv-u-v+2)\left(u^{2}v^{2}+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {3}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {4}{q^{17/2}}}-{\frac {3}{q^{19/2}}}+{\frac {3}{q^{21/2}}}-{\frac {2}{q^{23/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-za^{13}+z^{5}a^{11}+5z^{3}a^{11}+5za^{11}-z^{7}a^{9}-5z^{5}a^{9}-6z^{3}a^{9}-za^{9}+a^{9}z^{-1}-z^{7}a^{7}-6z^{5}a^{7}-11z^{3}a^{7}-7za^{7}-a^{7}z^{-1}$ (db)
Kauffman polynomial	$2a^{15}z+a^{14}z^{4}+a^{13}z^{7}-3a^{13}z^{5}+2a^{13}z^{3}+a^{13}z+2a^{12}z^{8}-10a^{12}z^{6}+17a^{12}z^{4}-11a^{12}z^{2}+a^{11}z^{9}-4a^{11}z^{7}+6a^{11}z^{5}-9a^{11}z^{3}+5a^{11}z+3a^{10}z^{8}-14a^{10}z^{6}+18a^{10}z^{4}-8a^{10}z^{2}+a^{9}z^{9}-4a^{9}z^{7}+3a^{9}z^{5}-a^{9}z+a^{9}z^{-1}+a^{8}z^{8}-4a^{8}z^{6}+2a^{8}z^{4}+3a^{8}z^{2}-a^{8}+a^{7}z^{7}-6a^{7}z^{5}+11a^{7}z^{3}-7a^{7}z+a^{7}z^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

0

-10

2

-12

1

0

-14

3

2

1

-16

1

0

-18

2

3

-1

-20

1

0

-22

1

2

-1

-24

2

Integral Khovanov Homology

(db, data source)

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