L11n204

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_7,16,8,17 X_11,18,12,19 X_2,19,3,20 X_3,12,4,13 X_20,13,21,14 X_14,5,15,6 X_6,9,7,10 X_15,22,16,9 X_17,8,18,1 X_21,4,22,5
Gauss code	{1, -4, -5, 11, 7, -8, -2, 10}, {8, -1, -3, 5, 6, -7, -9, 2, -10, 3, 4, -6, -11, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(t(1)t(2)^{2}+1\right)\left(t(1)^{2}t(2)^{3}+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{21/2}}}-{\frac {1}{q^{23/2}}}$ (db)
Signature	-8 (db)
HOMFLY-PT polynomial	$a^{13}\left(-z^{3}\right)-3a^{13}z-a^{13}z^{-1}+a^{11}z^{7}+8a^{11}z^{5}+20a^{11}z^{3}+17a^{11}z+3a^{11}z^{-1}-a^{9}z^{9}-9a^{9}z^{7}-28a^{9}z^{5}-36a^{9}z^{3}-18a^{9}z-2a^{9}z^{-1}$ (db)
Kauffman polynomial	$-za^{15}-a^{14}+z^{3}a^{13}-3za^{13}+a^{13}z^{-1}-z^{8}a^{12}+8z^{6}a^{12}-20z^{4}a^{12}+17z^{2}a^{12}-3a^{12}-z^{9}a^{11}+9z^{7}a^{11}-28z^{5}a^{11}+37z^{3}a^{11}-20za^{11}+3a^{11}z^{-1}-z^{8}a^{10}+8z^{6}a^{10}-20z^{4}a^{10}+17z^{2}a^{10}-3a^{10}-z^{9}a^{9}+9z^{7}a^{9}-28z^{5}a^{9}+36z^{3}a^{9}-18za^{9}+2a^{9}z^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-8

1

-10

1

-12

1

-14

1

-16

1

0

-18

1

0

-20

1

-1

-22

1

0

-24

1

Integral Khovanov Homology

(db, data source)

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