L11n133

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_3,10,4,11 X_2,17,3,18 X_12,5,13,6 X₆₇₁₈ X_9,16,10,17 X_13,20,14,21 X_15,22,16,7 X_19,4,20,5 X_21,14,22,15
Gauss code	{1, -4, -3, 10, 5, -6}, {6, -1, -7, 3, 2, -5, -8, 11, -9, 7, 4, -2, -10, 8, -11, 9}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{6}+uv^{4}-uv^{3}+uv^{2}+1}{uv^{3}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{25/2}}}-{\frac {1}{q^{23/2}}}+{\frac {1}{q^{21/2}}}-{\frac {1}{q^{19/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-z^{3}a^{13}-3za^{13}-2a^{13}z^{-1}+z^{7}a^{11}+8z^{5}a^{11}+20z^{3}a^{11}+18za^{11}+5a^{11}z^{-1}-z^{9}a^{9}-9z^{7}a^{9}-28z^{5}a^{9}-36z^{3}a^{9}-18za^{9}-3a^{9}z^{-1}$ (db)
Kauffman polynomial	$a^{16}z^{2}-a^{16}+a^{15}z^{3}-a^{15}z+a^{14}z^{2}-a^{13}z^{3}+4a^{13}z-2a^{13}z^{-1}+a^{12}z^{8}-8a^{12}z^{6}+20a^{12}z^{4}-18a^{12}z^{2}+5a^{12}+a^{11}z^{9}-9a^{11}z^{7}+28a^{11}z^{5}-38a^{11}z^{3}+23a^{11}z-5a^{11}z^{-1}+a^{10}z^{8}-8a^{10}z^{6}+20a^{10}z^{4}-18a^{10}z^{2}+5a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-36a^{9}z^{3}+18a^{9}z-3a^{9}z^{-1}$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-8

1

-10

1

-12

1

-14

1

-16

1

0

-18

2

1

-20

1

0

-22

1

0

-24

0

-26

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-10$	$i=-8$	$i=-6$
$r=-9$		${\mathbb {Z} }$
$r=-8$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-7$		${\mathbb {Z} }$
$r=-6$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-5$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-4$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }$