L10n86

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_13,20,14,15 X_16,8,17,7 X_8,16,9,15 X_11,18,12,19 X_19,12,20,13 X_17,14,18,5 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {5, -4, -8, 6, -7, 3}, {9, -1, 4, -5, 10, -2, -6, 7, -3, 8}

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

Polynomial invariants

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

χ

1

-1

1

-1

-3

2

1

-5

2

0

-7

3

1

2

-9

1

3

2

-11

3

2

1

-13

2

-15

1

2

-1

-17

1

Integral Khovanov Homology

(db, data source)

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