L11n100

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_20,13,21,14 X_7,17,8,16 X_19,9,20,8 X_9,19,10,18 X_17,11,18,10 X_22,15,5,16 X_14,21,15,22 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -4, 5, -6, 7, 11, -2, 3, -9, 8, 4, -7, 6, -5, -3, 9, -8}

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

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

1

-1

2

1

0

2

1

-1

-2

3

2

1

-4

2

3

1

-6

2

0

-8

1

2

1

-10

1

2

-1

-12

2

Integral Khovanov Homology

(db, data source)

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