L10n84

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(uv^{2}w^{2}+uv^{2}w+uv+vw^{3}+w^{2}+w\right)}{{\sqrt {u}}vw^{2}}}$ (db)
Jones polynomial	$q^{-12}-q^{-11}+2q^{-10}-q^{-9}+2q^{-8}-q^{-7}+q^{-6}+q^{-3}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{12}z^{-2}+a^{12}-2z^{2}a^{10}-2a^{10}z^{-2}-4a^{10}-z^{2}a^{8}+a^{8}z^{-2}+z^{6}a^{6}+6z^{4}a^{6}+9z^{2}a^{6}+3a^{6}$ (db)
Kauffman polynomial	$a^{14}z^{6}-5a^{14}z^{4}+6a^{14}z^{2}-2a^{14}+a^{13}z^{7}-4a^{13}z^{5}+2a^{13}z^{3}+a^{13}z+a^{12}z^{8}-5a^{12}z^{6}+7a^{12}z^{4}-6a^{12}z^{2}-a^{12}z^{-2}+4a^{12}+2a^{11}z^{7}-11a^{11}z^{5}+17a^{11}z^{3}-10a^{11}z+2a^{11}z^{-1}+a^{10}z^{8}-7a^{10}z^{6}+17a^{10}z^{4}-20a^{10}z^{2}-2a^{10}z^{-2}+10a^{10}+a^{9}z^{7}-7a^{9}z^{5}+14a^{9}z^{3}-10a^{9}z+2a^{9}z^{-1}-a^{8}z^{4}+a^{8}z^{2}-a^{8}z^{-2}+2a^{8}-a^{7}z^{3}+a^{7}z+a^{6}z^{6}-6a^{6}z^{4}+9a^{6}z^{2}-3a^{6}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-5

1

-7

1

-9

1

0

-11

1

-13

2

3

1

0

-15

1

-17

1

3

1

-19

1

-21

1

-23

1

0

-25

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-7$	$i=-5$	$i=-3$
$r=-10$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-9$		${\mathbb {Z} }$
$r=-8$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$