L10a90

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_20,12,9,11 X_14,6,15,5 X_2,9,3,10 X_4,14,5,13 X_6,20,7,19 X_16,7,17,8 X_18,15,19,16 X_8,17,1,18
Gauss code	{1, -5, 2, -6, 4, -7, 8, -10}, {5, -1, 3, -2, 6, -4, 9, -8, 10, -9, 7, -3}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)^{2}(t(2)-1)^{2}(t(1)t(2)+1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {1}{q^{9/2}}}+5q^{7/2}-{\frac {3}{q^{7/2}}}-8q^{5/2}+{\frac {5}{q^{5/2}}}+10q^{3/2}-{\frac {8}{q^{3/2}}}+q^{11/2}-11{\sqrt {q}}+{\frac {9}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-a^{3}z^{3}+3z^{3}a^{-3}-2a^{3}z+2za^{-3}-z^{7}a^{-1}+2az^{5}-5z^{5}a^{-1}+7az^{3}-9z^{3}a^{-1}+6az-6za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-3z^{8}a^{-2}-6z^{8}-3a^{3}z^{7}-5az^{7}-6z^{7}a^{-1}-4z^{7}a^{-3}-a^{4}z^{6}+7a^{2}z^{6}+z^{6}a^{-2}-4z^{6}a^{-4}+13z^{6}+10a^{3}z^{5}+22az^{5}+18z^{5}a^{-1}+3z^{5}a^{-3}-3z^{5}a^{-5}+3a^{4}z^{4}-a^{2}z^{4}+3z^{4}a^{-2}+4z^{4}a^{-4}-z^{4}a^{-6}-6z^{4}-9a^{3}z^{3}-22az^{3}-17z^{3}a^{-1}+4z^{3}a^{-5}-2a^{4}z^{2}-2a^{2}z^{2}-3z^{2}a^{-2}-z^{2}a^{-4}+z^{2}a^{-6}-z^{2}+3a^{3}z+8az+6za^{-1}-za^{-5}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

3

1

-2

6

5

2

3

4

5

3

-2

2

6

5

1

0

5

7

2

-2

3

4

-1

-4

2

5

3

-6

1

3

-2

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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