L10n26

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_16,11,17,12 X_7,15,8,14 X_15,9,16,8 X_20,17,5,18 X_18,14,19,13 X_12,20,13,19 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, -4, 5, 10, -2, 3, -8, 7, 4, -5, -3, 6, -7, 8, -6}

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

Polynomial invariants

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

χ

8

1

6

2

-2

4

3

1

2

4

2

-2

0

4

3

1

-2

4

5

1

-4

3

0

-6

1

4

3

-8

1

3

-2

-10

2

Integral Khovanov Homology

(db, data source)

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