L10n80

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Link Presentations

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

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

Polynomial invariants

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

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-1

2

-3

3

1

-2

-5

2

1

-7

3

0

-9

3

2

1

-11

3

5

2

-13

1

0

-15

3

-17

1

0

-19

1

Integral Khovanov Homology

(db, data source)

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