L11n156

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_11,21,12,20 X_5,12,6,13 X_19,4,20,5 X_14,18,15,17 X_16,8,17,7 X_22,16,7,15 X_18,14,19,13 X_6,21,1,22
Gauss code	{1, -2, 3, 6, -5, -11}, {8, -1, 2, -3, -4, 5, 10, -7, 9, -8, 7, -10, -6, 4, 11, -9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{3}-2u^{2}v^{2}+2u^{2}v+2uv^{4}-4uv^{3}+5uv^{2}-4uv+2u+2v^{3}-2v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	$2q^{5/2}-4q^{3/2}+6{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {9}{q^{3/2}}}-{\frac {9}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {5}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-2a^{3}z^{3}-a^{3}z-2az^{5}-7az^{3}+2z^{3}a^{-1}-8az-2az^{-1}+4za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-2a^{4}z^{8}-4a^{2}z^{8}-2z^{8}-3a^{5}z^{7}-a^{3}z^{7}+az^{7}-z^{7}a^{-1}-2a^{6}z^{6}+a^{4}z^{6}+9a^{2}z^{6}+6z^{6}-a^{7}z^{5}+7a^{5}z^{5}+2a^{3}z^{5}-7az^{5}-z^{5}a^{-1}+4a^{6}z^{4}+3a^{4}z^{4}-14a^{2}z^{4}-3z^{4}a^{-2}-16z^{4}+3a^{7}z^{3}-7a^{5}z^{3}-2a^{3}z^{3}+13az^{3}+5z^{3}a^{-1}-a^{6}z^{2}-3a^{4}z^{2}+10a^{2}z^{2}+6z^{2}a^{-2}+18z^{2}-2a^{7}z+5a^{5}z+a^{3}z-10az-4za^{-1}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

2

-2

4

2

4

2

-2

0

5

2

3

-2

5

0

-4

4

0

-6

3

5

2

-8

2

4

-2

-10

3

-12

1

2

-1

-14

1

Integral Khovanov Homology

(db, data source)

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