L11n218

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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_21,4,22,5 X_5,15,6,14 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 $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(u+1)^{2}(v-1)}{u^{3/2}{\sqrt {v}}}}$ (db)
Jones polynomial	$q^{13/2}-q^{11/2}+q^{9/2}-q^{7/2}-q^{5/2}-{\sqrt {q}}+{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{3}a^{-5}+3za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}-5z^{3}a^{-3}-6za^{-3}-3a^{-3}z^{-1}+az+2za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-3}-z^{9}a^{-5}-z^{8}a^{-2}-2z^{8}a^{-4}-z^{8}a^{-6}+7z^{7}a^{-3}+7z^{7}a^{-5}+7z^{6}a^{-2}+14z^{6}a^{-4}+7z^{6}a^{-6}-15z^{5}a^{-3}-15z^{5}a^{-5}-14z^{4}a^{-2}-28z^{4}a^{-4}-15z^{4}a^{-6}-z^{4}-az^{3}+z^{3}a^{-1}+15z^{3}a^{-3}+13z^{3}a^{-5}+10z^{2}a^{-2}+18z^{2}a^{-4}+11z^{2}a^{-6}+3z^{2}+2az-3za^{-1}-11za^{-3}-6za^{-5}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

14

1

-1

12

0

10

1

0

8

1

0

6

1

2

4

2

1

2

1

0

1

2

1

0

-2

0

-4

1

Integral Khovanov Homology

(db, data source)

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