L11n224

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_3,12,4,13 X_16,9,17,10 X_20,12,21,11 X_22,15,9,16 X_5,14,6,15 X_7,18,8,19 X_13,4,14,5 X_17,6,18,7 X_19,8,20,1 X_2,21,3,22
Gauss code	{1, -11, -2, 8, -6, 9, -7, 10}, {3, -1, 4, 2, -8, 6, 5, -3, -9, 7, -10, -4, 11, -5}

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

Polynomial invariants

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

-6

1

-8

1

-10

1

0

-12

2

-14

2

1

-1

-16

2

0

-18

1

2

1

0

-20

2

0

-22

1

2

1

-24

1

-1

-26

1

Integral Khovanov Homology

(db, data source)

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