L10n18

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_5,12,6,13 X₈₄₉₃ X_13,17,14,16 X_9,15,10,14 X_15,11,16,10 X_11,20,12,5 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {-4, -1, 2, -5, -7, 8, -9, 4, -6, 7, -8, 6, 10, -2, 3, 9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+2q^{5/2}-2q^{3/2}+{\sqrt {q}}-{\frac {2}{\sqrt {q}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{5}-z^{5}a^{-1}-a^{3}z^{3}+6az^{3}-5z^{3}a^{-1}+z^{3}a^{-3}-3a^{3}z+9az-8za^{-1}+2za^{-3}-a^{3}z^{-1}+4az^{-1}-4a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}-a^{3}z^{7}-2az^{7}-3z^{7}a^{-1}-2z^{7}a^{-3}-a^{4}z^{6}-a^{2}z^{6}+3z^{6}a^{-2}-z^{6}a^{-4}+4z^{6}+6a^{3}z^{5}+14az^{5}+17z^{5}a^{-1}+9z^{5}a^{-3}+5a^{4}z^{4}+8a^{2}z^{4}+4z^{4}a^{-2}+4z^{4}a^{-4}+3z^{4}-9a^{3}z^{3}-26az^{3}-26z^{3}a^{-1}-9z^{3}a^{-3}-5a^{4}z^{2}-12a^{2}z^{2}-10z^{2}a^{-2}-3z^{2}a^{-4}-14z^{2}+6a^{3}z+17az+15za^{-1}+4za^{-3}+a^{4}+4a^{2}+4a^{-2}+a^{-4}+7-a^{3}z^{-1}-4az^{-1}-4a^{-1}z^{-1}-a^{-3}z^{-1}$ (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

10

1

-1

8

1

6

1

0

4

1

2

1

0

2

1

0

1

4

2

1

-2

1

2

-4

1

0

-6

1

-8

0

-10

1

-1

Integral Khovanov Homology

(db, data source)

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