L10n24

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_13,18,14,19 X_9,17,10,16 X_17,9,18,8 X_15,20,16,5 X_19,14,20,15 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -9, 2, -10}, {9, -1, 3, 6, -5, -2, 10, -3, -4, 8, -7, 5, -6, 4, -8, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(v^{2}+1\right)}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+2q^{5/2}-3q^{3/2}+2{\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{-3}+2za^{-3}-z^{5}a^{-1}+az^{3}-4z^{3}a^{-1}+2az-4za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-4}-4z^{4}a^{-4}+3z^{2}a^{-4}+2z^{7}a^{-3}-9z^{5}a^{-3}+10z^{3}a^{-3}-4za^{-3}+z^{8}a^{-2}-3z^{6}a^{-2}-z^{4}a^{-2}+a^{2}z^{2}+3z^{2}a^{-2}+az^{7}+3z^{7}a^{-1}-5az^{5}-14z^{5}a^{-1}+8az^{3}+18z^{3}a^{-1}-4az-8za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{8}-4z^{6}+3z^{4}+z^{2}-1$ (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		\

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

1

6

1

0

4

2

1

2

1

0

3

2

1

-2

1

2

1

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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