L10n90

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,13,4,12 X_13,20,14,17 X_7,18,8,19 X_17,10,18,11 X_9,15,10,14 X_15,9,16,8 X_19,16,20,5 X₂₅₃₆ X_11,1,12,4
Gauss code	{1, -9, -2, 10}, {-5, 4, -8, 3}, {9, -1, -4, 7, -6, 5, -10, 2, -3, 6, -7, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(3)-1)\left(t(2)t(3)^{2}-2t(2)t(3)+2t(3)-1\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{3}-3q^{2}+5q-7+9q^{-1}-7q^{-2}+8q^{-3}-5q^{-4}+3q^{-5}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-z^{4}a^{4}-3z^{2}a^{4}-2a^{4}z^{-2}-5a^{4}+z^{6}a^{2}+4z^{4}a^{2}+7z^{2}a^{2}+a^{2}z^{-2}+6a^{2}-2z^{4}-5z^{2}-3+z^{2}a^{-2}+a^{-2}$ (db)
Kauffman polynomial	$2a^{2}z^{8}+2z^{8}+6a^{3}z^{7}+9az^{7}+3z^{7}a^{-1}+7a^{4}z^{6}+5a^{2}z^{6}+z^{6}a^{-2}-z^{6}+3a^{5}z^{5}-12a^{3}z^{5}-25az^{5}-10z^{5}a^{-1}-15a^{4}z^{4}-23a^{2}z^{4}-3z^{4}a^{-2}-11z^{4}+7a^{3}z^{3}+16az^{3}+9z^{3}a^{-1}+6a^{6}z^{2}+16a^{4}z^{2}+18a^{2}z^{2}+3z^{2}a^{-2}+11z^{2}+3a^{5}z+a^{3}z-3az-za^{-1}-4a^{6}-8a^{4}-7a^{2}-a^{-2}-3-2a^{5}z^{-1}-2a^{3}z^{-1}+a^{6}z^{-2}+2a^{4}z^{-2}+a^{2}z^{-2}$ (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

7

1

5

2

-2

3

1

2

1

4

2

-2

-1

5

3

2

-3

4

6

2

-5

4

3

1

-7

1

4

3

-9

2

4

-2

-11

3

Integral Khovanov Homology

(db, data source)

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