L10n61

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_16,7,17,8 X_5,1,6,10 X₃₇₄₆ X_9,5,10,4 X_20,17,11,18 X_18,13,19,14 X_14,19,15,20 X_2,11,3,12 X_8,15,9,16
Gauss code	{1, -9, -4, 5, -3, 4, 2, -10, -5, 3}, {9, -1, 7, -8, 10, -2, 6, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-2u^{3}v^{2}+2u^{3}v-u^{3}+2u^{2}v^{2}-2u^{2}v-2uv^{2}+2uv-v^{3}+2v^{2}-2v}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {7}{q^{5/2}}}+{\frac {5}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {2}{q^{13/2}}}+{\frac {4}{q^{11/2}}}+2{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{7}-2a^{7}z^{-1}+3z^{3}a^{5}+9za^{5}+7a^{5}z^{-1}-2z^{5}a^{3}-9z^{3}a^{3}-14za^{3}-7a^{3}z^{-1}+2z^{3}a+4za+2az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-4a^{8}z^{4}+5a^{8}z^{2}-2a^{8}+2a^{7}z^{7}-7a^{7}z^{5}+7a^{7}z^{3}-4a^{7}z+2a^{7}z^{-1}+a^{6}z^{8}+2a^{6}z^{6}-17a^{6}z^{4}+20a^{6}z^{2}-8a^{6}+6a^{5}z^{7}-20a^{5}z^{5}+22a^{5}z^{3}-17a^{5}z+7a^{5}z^{-1}+a^{4}z^{8}+5a^{4}z^{6}-24a^{4}z^{4}+29a^{4}z^{2}-13a^{4}+4a^{3}z^{7}-12a^{3}z^{5}+18a^{3}z^{3}-16a^{3}z+7a^{3}z^{-1}+4a^{2}z^{6}-11a^{2}z^{4}+17a^{2}z^{2}-8a^{2}+az^{5}+3az^{3}-3az+2az^{-1}+3z^{2}-2$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

χ

2

-2

0

2

-2

4

3

-1

-4

3

1

2

-6

2

4

2

-8

4

3

1

-10

1

3

2

-12

1

3

-2

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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