L10n27

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,17,12,16 X_14,7,15,8 X_8,15,9,16 X_17,5,18,20 X_13,18,14,19 X_19,12,20,13 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 $u$ , $v$ , $w$ , ...)	${\frac {uv^{3}-4uv^{2}+3uv-u-v^{3}+3v^{2}-4v+1}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {6}{q^{9/2}}}-{\frac {7}{q^{7/2}}}+{\frac {5}{q^{5/2}}}-{\frac {5}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {5}{q^{11/2}}}-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$za^{7}-2z^{3}a^{5}-2za^{5}+a^{5}z^{-1}+z^{5}a^{3}+2z^{3}a^{3}-a^{3}z^{-1}-z^{3}a-za$ (db)
Kauffman polynomial	$a^{9}z^{3}-a^{9}z+3a^{8}z^{4}-2a^{8}z^{2}+a^{7}z^{7}-a^{7}z^{5}+4a^{7}z^{3}-a^{7}z+a^{6}z^{8}-a^{6}z^{6}+3a^{6}z^{4}-a^{6}z^{2}+4a^{5}z^{7}-8a^{5}z^{5}+6a^{5}z^{3}-2a^{5}z+a^{5}z^{-1}+a^{4}z^{8}+2a^{4}z^{6}-7a^{4}z^{4}+3a^{4}z^{2}-a^{4}+3a^{3}z^{7}-6a^{3}z^{5}+a^{3}z^{3}-a^{3}z+a^{3}z^{-1}+3a^{2}z^{6}-7a^{2}z^{4}+2a^{2}z^{2}+az^{5}-2az^{3}+az$ (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

χ

2

1

0

2

-2

3

1

2

-4

3

0

-6

4

2

-8

2

3

1

-10

3

4

-1

-12

1

3

2

-14

2

-2

-16

1

Integral Khovanov Homology

(db, data source)

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