L10n89

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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_20,15,17,16 X_18,11,19,12 X_12,17,13,18 X_16,19,5,20 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 $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)\left(t(1)t(3)^{2}+t(2)t(3)^{2}-t(3)^{2}-t(1)t(3)+2t(1)t(2)t(3)-t(2)t(3)+2t(3)+t(1)-t(1)t(2)+t(2)\right)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{-10}-4q^{-9}+7q^{-8}-8q^{-7}+9q^{-6}-7q^{-5}+7q^{-4}-3q^{-3}+q^{-2}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{10}z^{-2}+2a^{10}-5z^{2}a^{8}-2a^{8}z^{-2}-8a^{8}+3z^{4}a^{6}+8z^{2}a^{6}+a^{6}z^{-2}+6a^{6}+z^{4}a^{4}+z^{2}a^{4}$ (db)
Kauffman polynomial	$3a^{12}z^{4}-5a^{12}z^{2}+2a^{12}+a^{11}z^{7}+a^{11}z^{5}-2a^{11}z^{3}+a^{10}z^{8}+2a^{10}z^{6}-3a^{10}z^{4}+a^{10}z^{-2}-2a^{10}+5a^{9}z^{7}-3a^{9}z^{5}-8a^{9}z^{3}+8a^{9}z-2a^{9}z^{-1}+a^{8}z^{8}+8a^{8}z^{6}-18a^{8}z^{4}+16a^{8}z^{2}+2a^{8}z^{-2}-9a^{8}+4a^{7}z^{7}-a^{7}z^{5}-8a^{7}z^{3}+8a^{7}z-2a^{7}z^{-1}+6a^{6}z^{6}-11a^{6}z^{4}+10a^{6}z^{2}+a^{6}z^{-2}-6a^{6}+3a^{5}z^{5}-2a^{5}z^{3}+a^{4}z^{4}-a^{4}z^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

1

-5

3

1

-2

-7

4

-9

3

0

-11

6

4

2

-13

4

5

1

-15

3

4

-1

-17

1

4

3

-19

1

3

-2

-21

2

Integral Khovanov Homology

(db, data source)

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