L11n69

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,16,8,17 X_22,18,5,17 X_18,12,19,11 X_20,14,21,13 X_12,20,13,19 X_14,22,15,21 X_15,8,16,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, 5, -7, 6, -8, -9, 3, 4, -5, 7, -6, 8, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

12

1

-1

10

2

8

3

1

-2

6

4

2

4

5

3

-2

2

4

0

4

6

2

-2

2

3

-1

-4

1

4

3

-6

1

2

-1

-8

2

Integral Khovanov Homology

(db, data source)

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