L10n87

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,19,12,18 X_17,9,18,8 X_7,17,8,16 X_13,15,14,20 X_15,5,16,14 X_19,13,20,12 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {-7, 5, -4, 3, -8, 6}, {9, -1, -5, 4, 10, -2, -3, 8, -6, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)\left(uvw^{2}+uw+u+v^{2}w^{3}+v^{2}w^{2}+vw\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{7}-q^{6}+2q^{5}-2q^{4}+2q^{3}-q^{2}+q+1+q^{-2}$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-z^{6}a^{-2}-7z^{4}a^{-2}+z^{4}-14z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}a^{-6}+5z^{2}-10a^{-2}+3a^{-4}+a^{-6}+6-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{4}a^{-8}-3z^{2}a^{-8}+a^{-8}+z^{5}a^{-7}-2z^{3}a^{-7}+z^{6}a^{-6}-3z^{4}a^{-6}+3z^{2}a^{-6}-a^{-6}+z^{5}a^{-5}-z^{3}a^{-5}+z^{4}a^{-4}-2z^{2}a^{-4}-a^{-4}z^{-2}+4a^{-4}+z^{7}a^{-3}-8z^{5}a^{-3}+19z^{3}a^{-3}-13za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}-9z^{6}a^{-2}+26z^{4}a^{-2}-30z^{2}a^{-2}-2a^{-2}z^{-2}+14a^{-2}+z^{7}a^{-1}-8z^{5}a^{-1}+18z^{3}a^{-1}-13za^{-1}+2a^{-1}z^{-1}+z^{8}-8z^{6}+21z^{4}-22z^{2}-z^{-2}+9$ (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

5

6

χ

15

1

13

1

0

11

1

9

1

0

7

1

2

1

0

5

1

2

1

3

1

2

1

0

1

2

-1

1

-3

1

-5

1

Integral Khovanov Homology

(db, data source)

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