L10n7

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_11,5,12,20 X_13,19,14,18 X_19,13,20,12 X_15,2,16,3
Gauss code	{1, 10, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 9, -8, 4, -10, -2, 3, 8, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {2uv^{3}-3uv^{2}-3v+2}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	${\frac {2}{q^{9/2}}}-{\frac {3}{q^{7/2}}}-q^{5/2}+{\frac {3}{q^{5/2}}}+q^{3/2}-{\frac {3}{q^{3/2}}}-{\frac {2}{q^{11/2}}}-2{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{-1}-z^{3}a^{5}-3za^{5}-a^{5}z^{-1}+z^{5}a^{3}+3z^{3}a^{3}-a^{3}z^{-1}+z^{5}a+4z^{3}a+4za+2az^{-1}-z^{3}a^{-1}-3za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$3a^{7}z-a^{7}z^{-1}+a^{6}z^{4}+a^{6}+2a^{5}z^{5}-4a^{5}z^{3}+3a^{5}z-a^{5}z^{-1}+2a^{4}z^{6}-4a^{4}z^{4}-a^{4}z^{2}+2a^{4}+2a^{3}z^{7}-7a^{3}z^{5}+8a^{3}z^{3}-7a^{3}z+a^{3}z^{-1}+a^{2}z^{8}-3a^{2}z^{6}+2a^{2}z^{4}-4a^{2}z^{2}+3a^{2}+3az^{7}+z^{7}a^{-1}-15az^{5}-6z^{5}a^{-1}+23az^{3}+11z^{3}a^{-1}-13az-6za^{-1}+2az^{-1}+a^{-1}z^{-1}+z^{8}-5z^{6}+7z^{4}-3z^{2}+1$ (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

6

1

4

0

2

1

0

1

-1

-2

2

0

-4

2

0

-6

1

0

-8

1

2

1

-10

1

0

-12

2

Integral Khovanov Homology

(db, data source)

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