L11n200

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,3,13,4 X_5,14,6,15 X_16,7,17,8 X_20,15,21,16 X_13,18,14,19 X_6,22,7,21 X_22,18,9,17 X_19,5,20,4 X_2,9,3,10 X_8,11,1,12
Gauss code	{1, -10, 2, 9, -3, -7, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, 8, 6, -9, -5, 7, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)+t(2))(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\frac {6}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {6}{q^{5/2}}}+{\frac {4}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {2}{q^{15/2}}}-{\frac {3}{q^{13/2}}}+{\frac {5}{q^{11/2}}}+{\sqrt {q}}-{\frac {3}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{7}+2za^{7}-z^{5}a^{5}-3z^{3}a^{5}-2za^{5}+a^{5}z^{-1}-z^{5}a^{3}-3z^{3}a^{3}-3za^{3}-a^{3}z^{-1}+z^{3}a+za$ (db)
Kauffman polynomial	$-z^{7}a^{9}+5z^{5}a^{9}-7z^{3}a^{9}+2za^{9}-2z^{8}a^{8}+10z^{6}a^{8}-15z^{4}a^{8}+7z^{2}a^{8}-z^{9}a^{7}+2z^{7}a^{7}+4z^{5}a^{7}-6z^{3}a^{7}-4z^{8}a^{6}+16z^{6}a^{6}-18z^{4}a^{6}+8z^{2}a^{6}-z^{9}a^{5}+z^{7}a^{5}+3z^{5}a^{5}+z^{3}a^{5}-za^{5}-a^{5}z^{-1}-2z^{8}a^{4}+5z^{6}a^{4}-3z^{4}a^{4}+z^{2}a^{4}+a^{4}-2z^{7}a^{3}+4z^{5}a^{3}-3z^{3}a^{3}+2za^{3}-a^{3}z^{-1}-z^{6}a^{2}-z^{2}a^{2}-3z^{3}a+za-z^{2}$ (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

1

χ

2

1

-1

0

2

-2

3

2

-1

-4

3

1

2

-6

2

3

1

-8

4

3

1

-10

2

3

1

-12

1

3

-2

-14

1

2

1

-16

1

-1

-18

1

Integral Khovanov Homology

(db, data source)

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