L11n8

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_17,1,18,4 X_9,14,10,15 X₃₈₄₉ X_5,11,6,10 X_11,20,12,21 X_19,22,20,5 X_13,19,14,18 X_21,12,22,13 X_2,16,3,15
Gauss code	{1, -11, -5, 3}, {-6, -1, 2, 5, -4, 6, -7, 10, -9, 4, 11, -2, -3, 9, -8, 7, -10, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)^{3}}{{\sqrt {u}}v^{3/2}}}$ (db)
Jones polynomial	$-q^{3/2}+2{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {2}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+2a^{5}z^{3}+5a^{5}z+3a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-7a^{3}z-3a^{3}z^{-1}+2az^{3}+4az+2az^{-1}-za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{5}z^{9}-a^{3}z^{9}-2a^{6}z^{8}-4a^{4}z^{8}-2a^{2}z^{8}-2a^{7}z^{7}+a^{5}z^{7}+2a^{3}z^{7}-az^{7}-a^{8}z^{6}+7a^{6}z^{6}+18a^{4}z^{6}+10a^{2}z^{6}+8a^{7}z^{5}+10a^{5}z^{5}+6a^{3}z^{5}+4az^{5}+4a^{8}z^{4}-3a^{6}z^{4}-24a^{4}z^{4}-19a^{2}z^{4}-2z^{4}-7a^{7}z^{3}-19a^{5}z^{3}-21a^{3}z^{3}-10az^{3}-z^{3}a^{-1}-3a^{8}z^{2}-3a^{6}z^{2}+8a^{4}z^{2}+11a^{2}z^{2}+3z^{2}+3a^{7}z+12a^{5}z+16a^{3}z+9az+2za^{-1}+a^{8}+2a^{6}-2a^{2}-a^{7}z^{-1}-3a^{5}z^{-1}-3a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

4

1

2

1

-1

0

3

1

2

-2

3

0

-4

3

2

1

2

-6

3

4

1

-8

2

3

-1

-10

1

3

1

-12

1

2

-1

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$	$i=0$
$r=-7$		${\mathbb {Z} }$
$r=-6$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$		${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$		${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=1$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=2$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$