L11n96

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_7,16,8,17 X_9,18,10,19 X_17,8,18,9 X_22,20,5,19 X_20,14,21,13 X_14,22,15,21 X_15,10,16,11 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, -3, 5, -4, 9, 11, -2, 7, -8, -9, 3, -5, 4, 6, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{5}-2uv^{4}+3uv^{3}-2uv^{2}+2uv-u-v^{5}+2v^{4}-2v^{3}+3v^{2}-2v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {7}{q^{5/2}}}-{\frac {8}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-8a^{3}z^{3}-5a^{3}z-a^{3}z^{-1}+az^{5}+3az^{3}+az$ (db)
Kauffman polynomial	$a^{9}z^{3}+3a^{8}z^{4}-2a^{8}z^{2}+5a^{7}z^{5}-6a^{7}z^{3}+2a^{7}z+a^{6}z^{8}-a^{6}z^{6}+4a^{6}z^{4}-3a^{6}z^{2}+a^{5}z^{9}-2a^{5}z^{7}+5a^{5}z^{5}-5a^{5}z^{3}-a^{5}z+a^{5}z^{-1}+4a^{4}z^{8}-12a^{4}z^{6}+11a^{4}z^{4}-2a^{4}z^{2}-a^{4}+a^{3}z^{9}+a^{3}z^{7}-11a^{3}z^{5}+11a^{3}z^{3}-4a^{3}z+a^{3}z^{-1}+3a^{2}z^{8}-10a^{2}z^{6}+7a^{2}z^{4}+3az^{7}-11az^{5}+9az^{3}-az+z^{6}-3z^{4}+z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

0

2

1

-1

-2

4

2

-4

4

3

-1

-6

4

3

1

-8

2

4

2

-10

3

4

-1

-12

1

3

2

-14

2

-2

-16

1

Integral Khovanov Homology

(db, data source)

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