L11n178

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X_13,21,14,20 X_3,10,4,11 X_5,14,6,15 X_7,16,8,17 X_15,22,16,7 X_11,4,12,5 X_19,13,20,12 X_21,1,22,6 X_2,18,3,17
Gauss code	{1, -11, -4, 8, -5, 10}, {-6, -1, 2, 4, -8, 9, -3, 5, -7, 6, 11, -2, -9, 3, -10, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {\left(uv^{2}-uv+u+v-1\right)\left(uv^{2}-uv-v^{2}+v-1\right)}{uv^{2}}}$ (db)
Jones polynomial	$-q^{3/2}+3{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {9}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}-a^{5}z^{5}+5a^{3}z^{5}-az^{5}-3a^{5}z^{3}+8a^{3}z^{3}-3az^{3}-2a^{5}z+4a^{3}z-3az+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-3a^{6}z^{8}-4a^{4}z^{8}-a^{2}z^{8}-3a^{7}z^{7}-2a^{5}z^{7}+a^{3}z^{7}-a^{8}z^{6}+8a^{6}z^{6}+8a^{4}z^{6}-a^{2}z^{6}+10a^{7}z^{5}+10a^{5}z^{5}-6a^{3}z^{5}-6az^{5}+3a^{8}z^{4}-3a^{6}z^{4}-3a^{4}z^{4}-3z^{4}-8a^{7}z^{3}-4a^{5}z^{3}+12a^{3}z^{3}+7az^{3}-z^{3}a^{-1}-2a^{8}z^{2}-a^{6}z^{2}+a^{4}z^{2}+a^{2}z^{2}+z^{2}+a^{7}z-a^{5}z-6a^{3}z-4az-a^{2}+a^{3}z^{-1}+az^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

4

1

2

-2

0

4

1

3

-2

4

3

-1

-4

5

3

2

-6

4

5

1

-8

3

4

-1

-10

2

4

2

-12

1

3

-2

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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