L10n94

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,10,19,9 X_6,18,1,17 X_7,17,8,16 X_3,10,4,11 X_14,6,15,5 X_4,14,5,13 X_11,13,12,20 X_15,7,16,12 X_19,3,20,2
Gauss code	{1, 10, -5, -7, 6, -3}, {-4, -1, 2, 5, -8, 9}, {7, -6, -9, 4, 3, -2, -10, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

0

1

2

3

4

5

6

7

8

χ

19

1

17

1

15

1

0

13

0

11

1

2

1

0

9

1

7

3

1

2

5

1

2

3

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$	$i=5$
$r=0$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=1$
$r=2$	${\mathbb {Z} }$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }$
$r=3$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=7$
$r=8$	${\mathbb {Z} }$	${\mathbb {Z} }$