L10n71

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,12,4,13 X_7,17,8,16 X_9,11,10,20 X_11,18,12,19 X_15,9,16,8 X_19,5,20,10 X_17,14,18,15 X₂₅₃₆ X_13,4,14,1
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 6, -4, 7}, {-5, 2, -10, 8, -6, 3, -8, 5, -7, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(3)^{2}t(2)^{2}+t(1)t(2)^{2}-2t(1)t(3)t(2)^{2}-t(2)^{2}-2t(1)t(3)^{2}t(2)+2t(1)t(3)t(2)-2t(3)t(2)+2t(2)+t(1)t(3)^{2}-t(3)^{2}+2t(3)-1}{{\sqrt {t(1)}}t(2)t(3)}}$ (db)
Jones polynomial	$q^{3}-2q^{2}+5q-4+7q^{-1}-6q^{-2}+5q^{-3}-4q^{-4}+2q^{-5}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}-a^{4}z^{4}-3a^{4}z^{2}-2a^{4}+a^{2}z^{6}+4a^{2}z^{4}+5a^{2}z^{2}+a^{2}z^{-2}+z^{2}a^{-2}+a^{-2}z^{-2}+3a^{2}+2a^{-2}-2z^{4}-6z^{2}-2z^{-2}-4$ (db)
Kauffman polynomial	$a^{2}z^{8}+z^{8}+3a^{3}z^{7}+5az^{7}+2z^{7}a^{-1}+3a^{4}z^{6}+4a^{2}z^{6}+z^{6}a^{-2}+2z^{6}+a^{5}z^{5}-6a^{3}z^{5}-13az^{5}-6z^{5}a^{-1}-5a^{4}z^{4}-17a^{2}z^{4}-4z^{4}a^{-2}-16z^{4}+3a^{5}z^{3}+7a^{3}z^{3}+6az^{3}+2z^{3}a^{-1}+3a^{6}z^{2}+6a^{4}z^{2}+14a^{2}z^{2}+6z^{2}a^{-2}+17z^{2}-2a^{5}z-6a^{3}z+4za^{-1}-a^{6}-a^{4}-3a^{2}-4a^{-2}-6-2az^{-1}-2a^{-1}z^{-1}+a^{2}z^{-2}+a^{-2}z^{-2}+2z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

1

-1

3

4

1

3

1

2

3

1

-1

5

2

3

-3

2

3

1

-5

3

4

-1

-7

1

2

1

-9

1

3

-2

-11

2

Integral Khovanov Homology

(db, data source)

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