L11n213

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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_14,5,15,6 X_9,18,10,19 X_17,22,18,9 X_21,1,22,8 X_20,15,21,16 X_7,16,8,17 X_4,13,5,14 X_6,20,7,19
Gauss code	{1, -2, 3, -10, 4, -11, -9, 7}, {-5, -1, 2, -3, 10, -4, 8, 9, -6, 5, 11, -8, -7, 6}

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

Polynomial invariants

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

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-4

2

-6

1

0

-8

3

1

2

-10

2

1

-1

-12

4

3

1

-14

3

0

-16

2

3

-1

-18

1

3

2

-20

1

2

-1

-22

1

-24

1

-1

Integral Khovanov Homology

(db, data source)

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