L11n254

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X_14,3,15,4 X_16,5,17,6 X_22,7,11,8 X_9,18,10,19 X_17,20,18,21 X_19,10,20,1 X_8,11,9,12 X_4,15,5,16 X_6,21,7,22
Gauss code	{1, -2, 3, -10, 4, -11, 5, -9, -6, 8}, {9, -1, 2, -3, 10, -4, -7, 6, -8, 7, 11, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{4}v^{4}+u^{3}v^{2}+u^{2}v^{3}-u^{2}v^{2}+u^{2}v+uv^{2}+1}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{17/2}}}-{\frac {1}{q^{19/2}}}+{\frac {2}{q^{21/2}}}-{\frac {2}{q^{23/2}}}+{\frac {1}{q^{25/2}}}-{\frac {1}{q^{27/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$-z^{3}a^{13}-2za^{13}+z^{7}a^{11}+8z^{5}a^{11}+19z^{3}a^{11}+14za^{11}+a^{11}z^{-1}-z^{9}a^{9}-9z^{7}a^{9}-28z^{5}a^{9}-36z^{3}a^{9}-17za^{9}-a^{9}z^{-1}$ (db)
Kauffman polynomial	$a^{17}z^{3}-2a^{17}z+a^{16}z^{4}-a^{16}z^{2}+a^{15}z^{5}-2a^{15}z^{3}+2a^{15}z+a^{14}z^{4}+2a^{13}z+a^{12}z^{8}-8a^{12}z^{6}+19a^{12}z^{4}-13a^{12}z^{2}+a^{11}z^{9}-9a^{11}z^{7}+27a^{11}z^{5}-33a^{11}z^{3}+15a^{11}z-a^{11}z^{-1}+a^{10}z^{8}-8a^{10}z^{6}+19a^{10}z^{4}-14a^{10}z^{2}+a^{10}+a^{9}z^{9}-9a^{9}z^{7}+28a^{9}z^{5}-36a^{9}z^{3}+17a^{9}z-a^{9}z^{-1}$ (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		\

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-8

1

-10

1

-12

1

-14

1

-16

1

-1

-18

1

0

-20

1

-1

-22

1

0

-24

1

-26

1

0

-28

1

Integral Khovanov Homology

(db, data source)

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