L11n76

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Planar diagram presentation	X₆₁₇₂ X_3,10,4,11 X_15,5,16,22 X_7,17,8,16 X_17,20,18,21 X_9,14,10,15 X_19,13,20,12 X_13,19,14,18 X_21,9,22,8 X₂₅₃₆ X_11,4,12,1
Gauss code	{1, -10, -2, 11}, {10, -1, -4, 9, -6, 2, -11, 7, -8, 6, -3, 4, -5, 8, -7, 5, -9, 3}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-5t(1)t(2)^{4}+t(2)^{4}+8t(1)t(2)^{3}-6t(2)^{3}-6t(1)t(2)^{2}+8t(2)^{2}+t(1)t(2)-5t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {3}{q^{9/2}}}-3q^{7/2}+{\frac {7}{q^{7/2}}}+7q^{5/2}-{\frac {11}{q^{5/2}}}-11q^{3/2}+{\frac {13}{q^{3/2}}}+13{\sqrt {q}}-{\frac {15}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}+z^{3}a^{-3}-4a^{3}z+2za^{-3}-2a^{3}z^{-1}+a^{-3}z^{-1}+az^{7}+4az^{5}-2z^{5}a^{-1}+6az^{3}-6z^{3}a^{-1}+5az-6za^{-1}+3az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2az^{9}-2z^{9}a^{-1}-7a^{2}z^{8}-4z^{8}a^{-2}-11z^{8}-8a^{3}z^{7}-11az^{7}-6z^{7}a^{-1}-3z^{7}a^{-3}-3a^{4}z^{6}+12a^{2}z^{6}+7z^{6}a^{-2}-z^{6}a^{-4}+23z^{6}+16a^{3}z^{5}+35az^{5}+27z^{5}a^{-1}+8z^{5}a^{-3}-3a^{4}z^{4}-13a^{2}z^{4}+2z^{4}a^{-2}+3z^{4}a^{-4}-11z^{4}-6a^{5}z^{3}-21a^{3}z^{3}-33az^{3}-25z^{3}a^{-1}-7z^{3}a^{-3}+3a^{4}z^{2}+9a^{2}z^{2}-6z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}+5a^{5}z+11a^{3}z+15az+12za^{-1}+3za^{-3}-2a^{2}+2a^{-2}+a^{-4}-a^{5}z^{-1}-2a^{3}z^{-1}-3az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

5

1

-4

4

6

2

4

2

7

5

-2

0

8

6

2

-2

6

8

2

-4

5

7

-2

-6

2

6

4

-8

1

5

-4

-10

3

Integral Khovanov Homology

(db, data source)

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