L10n88

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_15,17,16,20 X_11,19,12,18 X_17,13,18,12 X_19,5,20,16 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {-6, 5, -7, 4}, {9, -1, -3, 8, 10, -2, -5, 6, -8, 3, -4, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)\left(-t(1)t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}-t(1)t(3)-t(2)t(3)+t(1)-t(1)t(2)-t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$q^{3}+3-2q^{-1}+3q^{-2}-3q^{-3}+3q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}-2z^{2}a^{4}-2a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}+4a^{2}-z^{4}-6z^{2}-2z^{-2}-6+z^{2}a^{-2}+a^{-2}z^{-2}+3a^{-2}$ (db)
Kauffman polynomial	$a^{4}z^{8}+a^{2}z^{8}+2a^{5}z^{7}+3a^{3}z^{7}+az^{7}+a^{6}z^{6}-2a^{4}z^{6}-4a^{2}z^{6}+z^{6}a^{-2}-8a^{5}z^{5}-13a^{3}z^{5}-5az^{5}-4a^{6}z^{4}-4a^{4}z^{4}+3a^{2}z^{4}-6z^{4}a^{-2}-3z^{4}+7a^{5}z^{3}+15a^{3}z^{3}+5az^{3}-3z^{3}a^{-1}+4a^{6}z^{2}+4a^{4}z^{2}+a^{2}z^{2}+9z^{2}a^{-2}+10z^{2}-2a^{5}z-6a^{3}z+2az+6za^{-1}-a^{6}-2a^{2}-5a^{-2}-7-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

1

3

1

0

1

3

-1

2

4

1

-3

2

1

-5

1

2

1

0

-7

2

0

-9

1

2

1

-11

1

-1

-13

1

Integral Khovanov Homology

(db, data source)

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