L10a89

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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_18,16,19,15 X_16,7,17,8 X_6,17,7,18 X_8,20,1,19
Gauss code	{1, -5, 2, -6, 4, -9, 8, -10}, {5, -1, 3, -2, 6, -4, 7, -8, 9, -7, 10, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)\left(u^{2}v^{2}-u^{2}v-uv^{2}-u-v+1\right)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{15/2}+3q^{13/2}-4q^{11/2}+6q^{9/2}-8q^{7/2}+7q^{5/2}-7q^{3/2}+5{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-5}-3z^{3}a^{-5}-za^{-5}+z^{7}a^{-3}+5z^{5}a^{-3}+8z^{3}a^{-3}+5za^{-3}-2z^{5}a^{-1}+az^{3}-8z^{3}a^{-1}+3az-7za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-9}+3z^{4}a^{-8}-2z^{2}a^{-8}+4z^{5}a^{-7}-3z^{3}a^{-7}+za^{-7}+4z^{6}a^{-6}-2z^{4}a^{-6}-3z^{2}a^{-6}+4z^{7}a^{-5}-5z^{5}a^{-5}-2z^{3}a^{-5}+2za^{-5}+3z^{8}a^{-4}-5z^{6}a^{-4}-z^{4}a^{-4}+z^{2}a^{-4}+z^{9}a^{-3}+3z^{7}a^{-3}-18z^{5}a^{-3}+17z^{3}a^{-3}-4za^{-3}+5z^{8}a^{-2}-18z^{6}a^{-2}+15z^{4}a^{-2}-z^{2}a^{-2}+z^{9}a^{-1}+az^{7}-5az^{5}-14z^{5}a^{-1}+8az^{3}+23z^{3}a^{-1}-5az-10za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{8}-9z^{6}+11z^{4}-3z^{2}-1$ (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

6

χ

16

1

14

2

-2

12

2

1

10

4

2

-2

8

4

2

6

3

4

1

4

0

2

3

5

2

0

1

2

-1

-2

1

3

2

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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