L10n49

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_20,14,7,13 X_12,5,13,6 X_3,10,4,11 X_4,15,5,16 X_11,16,12,17 X_14,20,15,19 X_17,2,18,3
Gauss code	{1, 10, -6, -7, 5, -3}, {3, -1, 2, 6, -8, -5, 4, -9, 7, 8, -10, -2, 9, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-2u^{2}v^{3}-uv^{2}-2v+1}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{9/2}}}+{\frac {2}{q^{7/2}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$za^{9}+a^{9}z^{-1}-z^{5}a^{7}-5z^{3}a^{7}-6za^{7}-2a^{7}z^{-1}+z^{7}a^{5}+6z^{5}a^{5}+11z^{3}a^{5}+8za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-5z^{3}a^{3}-6za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{11}z+a^{10}z^{2}+2a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+a^{8}z^{6}-3a^{8}z^{4}+a^{8}z^{2}+2a^{7}z^{7}-10a^{7}z^{5}+15a^{7}z^{3}-10a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}-4a^{6}z^{6}+3a^{6}z^{4}-a^{6}z^{2}+a^{6}+3a^{5}z^{7}-16a^{5}z^{5}+24a^{5}z^{3}-12a^{5}z+2a^{5}z^{-1}+a^{4}z^{8}-5a^{4}z^{6}+6a^{4}z^{4}-a^{4}z^{2}+a^{3}z^{7}-6a^{3}z^{5}+11a^{3}z^{3}-7a^{3}z+a^{3}z^{-1}$ (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

0

1

-2

0

-4

2

1

-6

1

0

-8

1

0

-10

1

0

-12

1

0

-14

1

-16

1

0

-18

1

Integral Khovanov Homology

(db, data source)

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