L11n209

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)t(2)+1)(t(2)t(1)-t(1)+1)(t(1)t(2)-t(2)+1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {1}{\sqrt {q}}}-{\frac {3}{q^{3/2}}}+{\frac {3}{q^{5/2}}}-{\frac {6}{q^{7/2}}}+{\frac {6}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}z^{5}+4a^{7}z^{3}+4a^{7}z-a^{5}z^{7}-6a^{5}z^{5}-12a^{5}z^{3}-7a^{5}z+a^{5}z^{-1}+a^{3}z^{5}+3a^{3}z^{3}+a^{3}z-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{11}z^{5}-3a^{11}z^{3}+a^{11}z+2a^{10}z^{6}-6a^{10}z^{4}+3a^{10}z^{2}+2a^{9}z^{7}-5a^{9}z^{5}+2a^{9}z^{3}+2a^{8}z^{8}-7a^{8}z^{6}+10a^{8}z^{4}-5a^{8}z^{2}+a^{7}z^{9}-3a^{7}z^{7}+5a^{7}z^{5}-3a^{7}z^{3}+3a^{7}z+3a^{6}z^{8}-13a^{6}z^{6}+24a^{6}z^{4}-12a^{6}z^{2}+a^{5}z^{9}-5a^{5}z^{7}+14a^{5}z^{5}-15a^{5}z^{3}+6a^{5}z+a^{5}z^{-1}+a^{4}z^{8}-4a^{4}z^{6}+9a^{4}z^{4}-6a^{4}z^{2}-a^{4}+3a^{3}z^{5}-7a^{3}z^{3}+2a^{3}z+a^{3}z^{-1}+a^{2}z^{4}-2a^{2}z^{2}$ (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

0

1

-1

-2

2

-4

2

0

-6

4

1

3

-8

2

0

-10

4

0

-12

2

3

1

-14

1

3

-2

-16

1

2

1

-18

1

-1

-20

1

Integral Khovanov Homology

(db, data source)

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