L10n11

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_15,1,16,4 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_17,20,18,5 X_11,19,12,18 X_19,11,20,10 X_2,14,3,13
Gauss code	{1, -10, -5, 3}, {-4, -1, 2, 5, -6, 9, -8, 4, 10, -2, -3, 6, -7, 8, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{3}}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-2q^{3/2}+3{\sqrt {q}}-{\frac {5}{\sqrt {q}}}+{\frac {5}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}+a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-7za^{3}-3a^{3}z^{-1}+3z^{3}a+7za+4az^{-1}-2za^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+a^{7}z+2a^{6}z^{6}-6a^{6}z^{4}+4a^{6}z^{2}-a^{6}+2a^{5}z^{7}-5a^{5}z^{5}+3a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+a^{4}z^{8}-6a^{4}z^{4}+9a^{4}z^{2}-3a^{4}+4a^{3}z^{7}-13a^{3}z^{5}+20a^{3}z^{3}-13a^{3}z+3a^{3}z^{-1}+a^{2}z^{8}-a^{2}z^{6}+5a^{2}z^{2}-3a^{2}+2az^{7}-7az^{5}+17az^{3}+3z^{3}a^{-1}-15az-5za^{-1}+4az^{-1}+2a^{-1}z^{-1}+z^{6}-2$ (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

4

2

1

-1

0

4

2

-2

3

0

-4

3

2

1

-6

2

3

1

-8

1

3

-2

-10

1

2

1

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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