L11n15

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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_5,10,6,11 X₈₄₉₃ X_11,22,12,5 X_13,20,14,21 X_19,14,20,15 X_21,12,22,13 X_9,18,10,19 X_2,16,3,15
Gauss code	{1, -11, 5, -3}, {-4, -1, 2, -5, -10, 4, -6, 9, -7, 8, 11, -2, 3, 10, -8, 7, -9, 6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{5/2}}}+{\frac {1}{q^{19/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-za^{9}-a^{9}z^{-1}+z^{3}a^{7}+3za^{7}+2a^{7}z^{-1}-za^{5}-a^{5}z^{-1}+a^{3}z^{-1}-za-az^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{10}+7z^{6}a^{10}-15z^{4}a^{10}+11z^{2}a^{10}-3a^{10}-z^{9}a^{9}+7z^{7}a^{9}-15z^{5}a^{9}+11z^{3}a^{9}-3za^{9}+a^{9}z^{-1}-2z^{8}a^{8}+14z^{6}a^{8}-31z^{4}a^{8}+26z^{2}a^{8}-7a^{8}-z^{9}a^{7}+7z^{7}a^{7}-16z^{5}a^{7}+16z^{3}a^{7}-8za^{7}+2a^{7}z^{-1}-z^{8}a^{6}+7z^{6}a^{6}-16z^{4}a^{6}+15z^{2}a^{6}-4a^{6}-z^{5}a^{5}+5z^{3}a^{5}-5za^{5}+a^{5}z^{-1}-za^{3}+a^{3}z^{-1}-a^{2}-za+az^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

0

1

-2

1

2

1

-4

1

-6

1

2

1

-8

1

-10

1

2

1

0

-12

1

0

-14

1

0

-16

1

0

-18

0

-20

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$	$i=0$
$r=-9$		${\mathbb {Z} }$
$r=-8$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$		${\mathbb {Z} }$
$r=-6$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }$