L11n214

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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_9,18,10,19 X_17,22,18,9 X_21,1,22,8 X_20,13,21,14 X_5,14,6,15 X_7,16,8,17 X_15,6,16,7 X_4,20,5,19
Gauss code	{1, -2, 3, -11, -8, 10, -9, 6}, {-4, -1, 2, -3, 7, 8, -10, 9, -5, 4, 11, -7, -6, 5}

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

Polynomial invariants

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

2

1

-1

-8

5

1

4

-10

4

2

-2

-12

6

5

1

-14

5

0

-16

4

5

-1

-18

2

5

3

-20

1

4

-3

-22

2

-24

1

-1

Integral Khovanov Homology

(db, data source)

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