L10n5

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_13,5,14,20 X_11,19,12,18 X_19,13,20,12 X_2,16,3,15
Gauss code	{1, -10, -5, 3}, {-6, -1, 2, 5, -4, 6, -8, 9, -7, 4, 10, -2, -3, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$2q^{9/2}-2q^{7/2}+2q^{5/2}-{\frac {1}{q^{5/2}}}-3q^{3/2}+{\frac {1}{q^{3/2}}}-q^{11/2}+2{\sqrt {q}}-{\frac {2}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}+2z^{3}a^{-3}+3az-8za^{-1}+6za^{-3}-za^{-5}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}-az^{7}-3z^{7}a^{-1}-2z^{7}a^{-3}+3z^{6}a^{-2}-2z^{6}a^{-4}+5z^{6}+6az^{5}+16z^{5}a^{-1}+9z^{5}a^{-3}-z^{5}a^{-5}+2z^{4}a^{-2}+8z^{4}a^{-4}-6z^{4}-11az^{3}-26z^{3}a^{-1}-13z^{3}a^{-3}+2z^{3}a^{-5}-7z^{2}a^{-2}-9z^{2}a^{-4}-2z^{2}a^{-6}+7az+17za^{-1}+11za^{-3}-za^{-7}+3a^{-2}+3a^{-4}+a^{-6}+2-2az^{-1}-4a^{-1}z^{-1}-3a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

χ

12

1

10

1

-1

8

2

0

6

1

0

4

2

1

2

3

1

0

1

0

-2

1

2

1

-4

0

-6

1

Integral Khovanov Homology

(db, data source)

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