L11n220

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_8,9,1,10 X_3,12,4,13 X_15,22,16,9 X_2,17,3,18 X_21,4,22,5 X_14,5,15,6 X_20,13,21,14 X_11,16,12,17 X_6,19,7,20 X_18,7,19,8
Gauss code	{1, -5, -3, 6, 7, -10, 11, -2}, {2, -1, -9, 3, 8, -7, -4, 9, 5, -11, 10, -8, -6, 4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{5}+t(1)^{3}t(2)^{4}-t(1)^{2}t(2)^{4}+t(1)^{2}t(2)^{3}+t(1)t(2)^{2}-t(1)t(2)+t(2)+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {2}{q^{11/2}}}+{\frac {2}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {2}{q^{19/2}}}+{\frac {2}{q^{21/2}}}-{\frac {1}{q^{23/2}}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$a^{13}(-z)-a^{13}z^{-1}+a^{11}z^{5}+6a^{11}z^{3}+9a^{11}z+3a^{11}z^{-1}-a^{9}z^{7}-6a^{9}z^{5}-11a^{9}z^{3}-8a^{9}z-2a^{9}z^{-1}-a^{7}z^{7}-6a^{7}z^{5}-10a^{7}z^{3}-4a^{7}z$ (db)
Kauffman polynomial	$a^{15}z+a^{14}z^{2}+a^{14}+a^{13}z^{5}-3a^{13}z^{3}+3a^{13}z-a^{13}z^{-1}+a^{12}z^{8}-6a^{12}z^{6}+13a^{12}z^{4}-12a^{12}z^{2}+3a^{12}+a^{11}z^{9}-7a^{11}z^{7}+20a^{11}z^{5}-29a^{11}z^{3}+16a^{11}z-3a^{11}z^{-1}+2a^{10}z^{8}-11a^{10}z^{6}+19a^{10}z^{4}-14a^{10}z^{2}+3a^{10}+a^{9}z^{9}-6a^{9}z^{7}+13a^{9}z^{5}-16a^{9}z^{3}+10a^{9}z-2a^{9}z^{-1}+a^{8}z^{8}-5a^{8}z^{6}+6a^{8}z^{4}-a^{8}z^{2}+a^{7}z^{7}-6a^{7}z^{5}+10a^{7}z^{3}-4a^{7}z$ (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

-6

1

-8

1

0

-10

1

-12

1

0

-14

2

1

-16

1

-18

2

0

-20

0

-22

1

2

-1

-24

1

Integral Khovanov Homology

(db, data source)

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