L10n101

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_18,12,19,11 X_20,16,17,15 X_16,20,9,19 X_12,18,13,17 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 4}, {8, -5, 7, -6}, {-10, 2, 5, -8, -4, 3, 6, -7}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(4)-1)\left(-t(4)^{2}+t(1)t(4)+t(2)t(4)-t(1)t(3)t(4)+t(1)t(2)t(3)t(4)-t(2)t(3)t(4)-t(4)+t(1)t(2)t(3)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}t(4)^{3/2}}}$ (db)
Jones polynomial	$2q^{9/2}-4q^{7/2}+2q^{5/2}-4q^{3/2}-q^{17/2}-q^{13/2}-3q^{11/2}+2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-a^{-9}z^{-3}-a^{-9}z^{-1}+3a^{-7}z^{-3}+2za^{-7}+5a^{-7}z^{-1}-3a^{-5}z^{-3}-3za^{-5}-6a^{-5}z^{-1}-z^{5}a^{-3}-3z^{3}a^{-3}+a^{-3}z^{-3}-za^{-3}+a^{-3}z^{-1}+z^{3}a^{-1}+2za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{7}a^{-9}-7z^{5}a^{-9}+15z^{3}a^{-9}-a^{-9}z^{-3}-14za^{-9}+6a^{-9}z^{-1}+z^{6}a^{-8}-9z^{4}a^{-8}+19z^{2}a^{-8}+3a^{-8}z^{-2}-13a^{-8}+z^{7}a^{-7}-10z^{5}a^{-7}+28z^{3}a^{-7}-3a^{-7}z^{-3}-27za^{-7}+14a^{-7}z^{-1}+2z^{6}a^{-6}-14z^{4}a^{-6}+33z^{2}a^{-6}+6a^{-6}z^{-2}-24a^{-6}+z^{7}a^{-5}-4z^{5}a^{-5}+10z^{3}a^{-5}-3a^{-5}z^{-3}-16za^{-5}+12a^{-5}z^{-1}+3z^{6}a^{-4}-10z^{4}a^{-4}+14z^{2}a^{-4}+3a^{-4}z^{-2}-11a^{-4}+z^{7}a^{-3}-6z^{3}a^{-3}-a^{-3}z^{-3}+3a^{-3}z^{-1}+2z^{6}a^{-2}-5z^{4}a^{-2}+a^{-2}+z^{5}a^{-1}-3z^{3}a^{-1}+3za^{-1}-a^{-1}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

χ

18

1

16

1

14

1

12

4

10

3

5

1

8

2

6

3

1

2

4

2

1

3

2

0

1

-1

-2

1

Integral Khovanov Homology

(db, data source)

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