L10n92

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,10,19,9 X_17,6,18,1 X_7,17,8,16 X_3,10,4,11 X_5,14,6,15 X_13,4,14,5 X_11,13,12,20 X_15,7,16,12 X_2,19,3,20
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 $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)^{2}t(3)^{3}-t(1)t(2)^{2}t(3)^{2}+t(2)^{2}t(3)^{2}+t(1)t(2)t(3)^{2}-t(1)^{2}t(3)+t(1)t(3)-t(1)t(2)t(3)-t(1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{-6}-q^{-5}+2q^{-4}-2q^{-3}+q^{2}+3q^{-2}-q-2q^{-1}+3$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+4a^{4}z^{2}+a^{4}z^{-2}+4a^{4}-a^{2}z^{6}-6a^{2}z^{4}-12a^{2}z^{2}-2a^{2}z^{-2}-9a^{2}+z^{4}+4z^{2}+z^{-2}+5$ (db)
Kauffman polynomial	$a^{6}z^{6}-5a^{6}z^{4}+6a^{6}z^{2}-a^{6}+a^{5}z^{7}-4a^{5}z^{5}+3a^{5}z^{3}+a^{4}z^{8}-5a^{4}z^{6}+9a^{4}z^{4}-10a^{4}z^{2}-a^{4}z^{-2}+5a^{4}+2a^{3}z^{7}-9a^{3}z^{5}+13a^{3}z^{3}-9a^{3}z+2a^{3}z^{-1}+a^{2}z^{8}-6a^{2}z^{6}+16a^{2}z^{4}-21a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+11a^{2}-a^{-2}+az^{7}-5az^{5}+11az^{3}+z^{3}a^{-1}-9az+2az^{-1}+2z^{4}-4z^{2}-z^{-2}+5$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

χ

5

1

3

1

0

1

2

-1

2

3

1

-3

1

-5

1

2

1

-7

1

0

-9

1

-11

1

0

-13

1

Integral Khovanov Homology

(db, data source)

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