L11n79

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,9,17,10 X_4,21,1,22 X_11,14,12,15 X_3,10,4,11 X_5,13,6,12 X_13,5,14,22 X_15,2,16,3 X_20,18,21,17 X_18,8,19,7 X_8,20,9,19
Gauss code	{1, 8, -5, -3}, {-6, -1, 10, -11, 2, 5, -4, 6, -7, 4, -8, -2, 9, -10, 11, -9, 3, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-4uv^{4}+6uv^{3}-4uv^{2}-4v^{3}+6v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {2}{q^{9/2}}}-3q^{7/2}+{\frac {4}{q^{7/2}}}+6q^{5/2}-{\frac {7}{q^{5/2}}}-8q^{3/2}+{\frac {9}{q^{3/2}}}+9{\sqrt {q}}-{\frac {11}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z+a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}+z^{3}a^{-3}-5a^{3}z+2za^{-3}-2a^{3}z^{-1}+a^{-3}z^{-1}+az^{7}+5az^{5}-2z^{5}a^{-1}+9az^{3}-7z^{3}a^{-1}+7az-7za^{-1}+3az^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-3z^{8}a^{-2}-6z^{8}-3a^{3}z^{7}-6az^{7}-6z^{7}a^{-1}-3z^{7}a^{-3}-a^{4}z^{6}+6a^{2}z^{6}+5z^{6}a^{-2}-z^{6}a^{-4}+13z^{6}+7a^{3}z^{5}+24az^{5}+26z^{5}a^{-1}+9z^{5}a^{-3}-2a^{4}z^{4}-10a^{2}z^{4}+6z^{4}a^{-2}+3z^{4}a^{-4}-5z^{4}-3a^{5}z^{3}-15a^{3}z^{3}-32az^{3}-27z^{3}a^{-1}-7z^{3}a^{-3}+2a^{4}z^{2}+7a^{2}z^{2}-8z^{2}a^{-2}-3z^{2}a^{-4}+4a^{5}z+10a^{3}z+16az+13za^{-1}+3za^{-3}-2a^{2}+2a^{-2}+a^{-4}-a^{5}z^{-1}-2a^{3}z^{-1}-3az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

4

1

-3

4

2

5

4

-1

0

6

4

2

-2

4

6

2

-4

3

5

-2

-6

1

4

3

-8

1

3

-2

-10

2

Integral Khovanov Homology

(db, data source)

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