L11n36

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_11,14,12,15 X₃₈₄₉ X_5,13,6,12 X_13,5,14,22 X_15,19,16,18 X_9,17,10,16 X_17,11,18,10 X_19,2,20,3
Gauss code	{1, 11, -5, -3}, {-6, -1, 2, 5, -9, 10, -4, 6, -7, 4, -8, 9, -10, 8, -11, -2, 3, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-4uv^{4}+5uv^{3}-3uv^{2}-3v^{3}+5v^{2}-4v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+4q^{5/2}-7q^{3/2}+8{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {8}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {2}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-a^{3}z^{5}+5az^{5}-2z^{5}a^{-1}-4a^{3}z^{3}+9az^{3}-8z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-5a^{3}z+8az-9za^{-1}+3za^{-3}+a^{5}z^{-1}-2a^{3}z^{-1}+3az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$3a^{5}z^{3}-4a^{5}z+a^{5}z^{-1}+a^{4}z^{6}+z^{6}a^{-4}+2a^{4}z^{4}-4z^{4}a^{-4}-2a^{4}z^{2}+4z^{2}a^{-4}-a^{-4}+3a^{3}z^{7}+2z^{7}a^{-3}-7a^{3}z^{5}-7z^{5}a^{-3}+13a^{3}z^{3}+7z^{3}a^{-3}-9a^{3}z-4za^{-3}+2a^{3}z^{-1}+a^{-3}z^{-1}+3a^{2}z^{8}+2z^{8}a^{-2}-8a^{2}z^{6}-4z^{6}a^{-2}+14a^{2}z^{4}-3z^{4}a^{-2}-9a^{2}z^{2}+6z^{2}a^{-2}+2a^{2}-2a^{-2}+az^{9}+z^{9}a^{-1}+3az^{7}+2z^{7}a^{-1}-15az^{5}-15z^{5}a^{-1}+24az^{3}+21z^{3}a^{-1}-15az-14za^{-1}+3az^{-1}+3a^{-1}z^{-1}+5z^{8}-14z^{6}+13z^{4}-5z^{2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

1

6

3

1

-2

4

1

3

2

4

3

-1

0

5

4

1

-2

4

5

1

-4

3

4

-1

-6

1

4

3

-8

1

3

-2

-10

2

Integral Khovanov Homology

(db, data source)

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