L11n90

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_7,16,8,17 X_17,22,18,5 X_14,9,15,10 X_10,20,11,19 X_21,9,22,8 X_18,14,19,13 X_20,15,21,16 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 7, 5, -6, 11, -2, 8, -5, 9, 3, -4, -8, 6, -9, -7, 4}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(2)-2)(t(1)+t(2))(2t(2)-1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {3}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {5}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {2}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-3za^{7}-2a^{7}z^{-1}+2z^{3}a^{5}+2za^{5}+a^{5}z^{-1}+2z^{3}a^{3}+2za^{3}+a^{3}z^{-1}-3za-az^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{9}+5z^{5}a^{9}-8z^{3}a^{9}+5za^{9}-a^{9}z^{-1}-2z^{8}a^{8}+9z^{6}a^{8}-11z^{4}a^{8}+4z^{2}a^{8}-a^{8}-z^{9}a^{7}+15z^{5}a^{7}-25z^{3}a^{7}+13za^{7}-2a^{7}z^{-1}-5z^{8}a^{6}+21z^{6}a^{6}-25z^{4}a^{6}+13z^{2}a^{6}-3a^{6}-z^{9}a^{5}-z^{7}a^{5}+15z^{5}a^{5}-19z^{3}a^{5}+9za^{5}-a^{5}z^{-1}-3z^{8}a^{4}+12z^{6}a^{4}-17z^{4}a^{4}+12z^{2}a^{4}-2a^{4}-2z^{7}a^{3}+5z^{5}a^{3}-2z^{3}a^{3}-3za^{3}+a^{3}z^{-1}-3z^{4}a^{2}+3z^{2}a^{2}-a^{2}-4za+az^{-1}$ (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

0

3

-2

2

1

-1

-4

3

2

1

-6

3

2

-1

-8

3

0

-10

3

4

1

-12

1

2

-1

-14

1

3

2

-16

1

-1

-18

1

Integral Khovanov Homology

(db, data source)

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