L11n212

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Planar diagram presentation	X_10,1,11,2 X_13,21,14,20 X_3,12,4,13 X_2,19,3,20 X_14,5,15,6 X_16,7,17,8 X_8,9,1,10 X_18,12,19,11 X_6,15,7,16 X_22,18,9,17 X_21,4,22,5
Gauss code	{1, -4, -3, 11, 5, -9, 6, -7}, {7, -1, 8, 3, -2, -5, 9, -6, 10, -8, 4, 2, -11, -10}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(uv+1)\left(u^{2}v+uv^{4}-2uv^{3}+2uv^{2}-2uv+u+v^{3}\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\frac {1}{\sqrt {q}}}-{\frac {3}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {7}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+4a^{7}z^{3}+5a^{7}z+2a^{7}z^{-1}-a^{5}z^{7}-6a^{5}z^{5}-13a^{5}z^{3}-11a^{5}z-3a^{5}z^{-1}+a^{3}z^{5}+3a^{3}z^{3}+2a^{3}z+a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{11}z^{5}-3a^{11}z^{3}+a^{11}z+2a^{10}z^{6}-5a^{10}z^{4}+a^{10}z^{2}+3a^{9}z^{7}-10a^{9}z^{5}+11a^{9}z^{3}-5a^{9}z+2a^{8}z^{8}-5a^{8}z^{6}+4a^{8}z^{4}+a^{7}z^{9}-2a^{7}z^{7}+3a^{7}z^{5}-3a^{7}z^{3}+6a^{7}z-2a^{7}z^{-1}+3a^{6}z^{8}-11a^{6}z^{6}+20a^{6}z^{4}-12a^{6}z^{2}+3a^{6}+a^{5}z^{9}-5a^{5}z^{7}+17a^{5}z^{5}-23a^{5}z^{3}+14a^{5}z-3a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+12a^{4}z^{4}-13a^{4}z^{2}+3a^{4}+3a^{3}z^{5}-6a^{3}z^{3}+2a^{3}z-a^{3}z^{-1}+a^{2}z^{4}-2a^{2}z^{2}+a^{2}$ (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

χ

0

1

-1

-2

2

-4

3

2

-1

-6

3

1

2

-8

4

3

-1

-10

3

0

-12

2

4

2

-14

2

3

-1

-16

2

-18

1

2

-1

-20

1

Integral Khovanov Homology

(db, data source)

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