L10n100

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_11,18,12,19 X_20,16,17,15 X_16,20,9,19 X_17,12,18,13 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -3, 4}, {-8, 5, 7, -6}, {-10, 2, -5, 8, -4, 3, 6, -7}

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

Polynomial invariants

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

-3

-2

-1

0

1

2

3

4

χ

10

2

8

3

2

-1

6

4

3

2

7

4

3

0

3

6

3

-2

1

0

-4

3

-6

1

-1

Integral Khovanov Homology

(db, data source)

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