L10a33

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_16,8,17,7 X_18,10,19,9 X_20,13,5,14 X_8,18,9,17 X_14,19,15,20 X_10,16,11,15 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -6, 4, -8, 10, -2, 5, -7, 8, -3, 6, -4, 7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2uv^{4}-4uv^{3}+4uv^{2}-4uv+u+v^{5}-4v^{4}+4v^{3}-4v^{2}+2v}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$-q^{9/2}+{\frac {2}{q^{9/2}}}+3q^{7/2}-{\frac {5}{q^{7/2}}}-6q^{5/2}+{\frac {6}{q^{5/2}}}+8q^{3/2}-{\frac {9}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-9{\sqrt {q}}+{\frac {10}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z+2a^{5}z^{-1}-3a^{3}z^{3}-z^{3}a^{-3}-8a^{3}z-4a^{3}z^{-1}-za^{-3}+2az^{5}+z^{5}a^{-1}+7az^{3}+z^{3}a^{-1}+8az+3az^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-6a^{2}z^{8}-4z^{8}-a^{5}z^{7}-2a^{3}z^{7}-8az^{7}-7z^{7}a^{-1}+8a^{4}z^{6}+18a^{2}z^{6}-8z^{6}a^{-2}+2z^{6}+5a^{5}z^{5}+21a^{3}z^{5}+33az^{5}+11z^{5}a^{-1}-6z^{5}a^{-3}-8a^{4}z^{4}-10a^{2}z^{4}+12z^{4}a^{-2}-3z^{4}a^{-4}+13z^{4}-9a^{5}z^{3}-33a^{3}z^{3}-33az^{3}-3z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}-a^{4}z^{2}-4a^{2}z^{2}-6z^{2}a^{-2}-9z^{2}+7a^{5}z+17a^{3}z+14az+2za^{-1}-2za^{-3}+2a^{4}+3a^{2}+a^{-2}+3-2a^{5}z^{-1}-4a^{3}z^{-1}-3az^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

10

1

8

2

-2

6

4

1

3

4

2

-2

2

5

4

1

0

6

5

-1

-2

3

4

-1

-4

3

6

3

-6

2

3

-1

-8

1

4

3

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

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