L11n219

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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_22,16,9,15 X_17,3,18,2 X_4,22,5,21 X_14,5,15,6 X_13,21,14,20 X_16,12,17,11 X_19,7,20,6 X_7,19,8,18
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 $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)\left(t(1)^{2}+t(1)+1\right)(t(2)-1)}{t(1)^{3/2}{\sqrt {t(2)}}}}$ (db)
Jones polynomial	$q^{13/2}-q^{11/2}+q^{9/2}-2q^{7/2}+q^{5/2}-q^{3/2}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{-5}+3za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}-5z^{3}a^{-3}-7za^{-3}-3a^{-3}z^{-1}+z^{3}a^{-1}+4za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{8}a^{-6}-7z^{6}a^{-6}+15z^{4}a^{-6}-10z^{2}a^{-6}+a^{-6}+z^{9}a^{-5}-7z^{7}a^{-5}+16z^{5}a^{-5}-15z^{3}a^{-5}+6za^{-5}-a^{-5}z^{-1}+2z^{8}a^{-4}-13z^{6}a^{-4}+25z^{4}a^{-4}-16z^{2}a^{-4}+3a^{-4}+z^{9}a^{-3}-7z^{7}a^{-3}+17z^{5}a^{-3}-21z^{3}a^{-3}+13za^{-3}-3a^{-3}z^{-1}+z^{8}a^{-2}-6z^{6}a^{-2}+10z^{4}a^{-2}-7z^{2}a^{-2}+3a^{-2}+z^{5}a^{-1}-6z^{3}a^{-1}+7za^{-1}-2a^{-1}z^{-1}-z^{2}$ (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		\

0

1

2

3

4

5

6

7

χ

14

1

-1

12

0

10

1

0

8

1

6

1

4

1

0

2

1

0

2

1

-2

1

Integral Khovanov Homology

(db, data source)

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