L11n236

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Planar diagram presentation	X_10,1,11,2 X_2,11,3,12 X_14,3,15,4 X_20,13,21,14 X_12,21,13,22 X_22,5,9,6 X_7,16,8,17 X_15,18,16,19 X_17,8,18,1 X_6,9,7,10 X_4,19,5,20
Gauss code	{1, -2, 3, -11, 6, -10, -7, 9}, {10, -1, 2, -5, 4, -3, -8, 7, -9, 8, 11, -4, 5, -6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(2u^{2}v-u^{2}-v+2\right)\left(uv^{2}+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {1}{q^{19/2}}}-{\frac {1}{q^{21/2}}}+{\frac {1}{q^{23/2}}}-{\frac {1}{q^{25/2}}}$ (db)
Signature	-6 (db)
HOMFLY-PT polynomial	$a^{13}(-z)+a^{11}z^{5}+6a^{11}z^{3}+6a^{11}z-a^{9}z^{7}-6a^{9}z^{5}-9a^{9}z^{3}-3a^{9}z+a^{9}z^{-1}-a^{7}z^{7}-6a^{7}z^{5}-10a^{7}z^{3}-6a^{7}z-a^{7}z^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{15}+4z^{3}a^{15}-3za^{15}-z^{6}a^{14}+4z^{4}a^{14}-2z^{2}a^{14}-z^{5}a^{13}+5z^{3}a^{13}-3za^{13}-z^{8}a^{12}+7z^{6}a^{12}-14z^{4}a^{12}+10z^{2}a^{12}-z^{9}a^{11}+7z^{7}a^{11}-15z^{5}a^{11}+15z^{3}a^{11}-7za^{11}-2z^{8}a^{10}+13z^{6}a^{10}-22z^{4}a^{10}+10z^{2}a^{10}-z^{9}a^{9}+6z^{7}a^{9}-9z^{5}a^{9}+4z^{3}a^{9}-za^{9}-a^{9}z^{-1}-z^{8}a^{8}+5z^{6}a^{8}-4z^{4}a^{8}-2z^{2}a^{8}+a^{8}-z^{7}a^{7}+6z^{5}a^{7}-10z^{3}a^{7}+6za^{7}-a^{7}z^{-1}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-6

1

-8

1

0

-10

1

-12

2

1

2

-14

1

2

1

0

-16

1

2

1

0

-18

1

2

-1

-20

1

0

-22

1

2

1

0

-24

0

-26

1

Integral Khovanov Homology

(db, data source)

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