L10n9

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₈₄₉₃ X_5,11,6,10 X_11,5,12,20 X_13,19,14,18 X_19,13,20,12 X_2,16,3,15
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -4, 6, -7, 9, -8, 4, 10, -2, 3, 8, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-q^{11/2}+q^{9/2}-q^{7/2}+q^{5/2}-q^{3/2}+{\sqrt {q}}-{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{-5}-a^{-5}z^{-1}+z^{3}a^{-3}+a^{3}z^{-1}+3za^{-3}+2a^{-3}z^{-1}-az-az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}a^{-4}-z^{7}a^{-1}-2z^{7}a^{-3}-z^{7}a^{-5}+6z^{6}a^{-2}+6z^{6}a^{-4}+6z^{5}a^{-1}+12z^{5}a^{-3}+6z^{5}a^{-5}-11z^{4}a^{-2}-10z^{4}a^{-4}-z^{4}-11z^{3}a^{-1}-21z^{3}a^{-3}-10z^{3}a^{-5}+9z^{2}a^{-2}+5z^{2}a^{-4}+4z^{2}-a^{3}z-az+8za^{-1}+13za^{-3}+5za^{-5}-a^{2}-3a^{-2}-a^{-4}-2+a^{3}z^{-1}+az^{-1}-a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

0

8

1

0

6

1

0

4

1

0

2

1

2

1

0

1

2

1

0

-2

1

2

1

-4

1

-6

1

Integral Khovanov Homology

(db, data source)

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