L11n206

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_16,8,17,7 X_18,12,19,11 X_19,3,20,2 X_3,12,4,13 X_13,21,14,20 X_14,5,15,6 X_6,9,7,10 X_22,16,9,15 X_8,18,1,17 X_4,22,5,21
Gauss code	{1, 4, -5, -11, 7, -8, 2, -10}, {8, -1, 3, 5, -6, -7, 9, -2, 10, -3, -4, 6, 11, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(1)t(2)+1)^{2}}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+5q^{7/2}-6q^{5/2}+4q^{3/2}-{\frac {1}{q^{3/2}}}+q^{15/2}-2q^{13/2}+3q^{11/2}-4{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-5}+4z^{3}a^{-5}+4za^{-5}+a^{-5}z^{-1}-z^{7}a^{-3}-6z^{5}a^{-3}-12z^{3}a^{-3}-9za^{-3}-3a^{-3}z^{-1}+z^{5}a^{-1}+4z^{3}a^{-1}+5za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-4z^{4}a^{-8}+3z^{2}a^{-8}+2z^{7}a^{-7}-8z^{5}a^{-7}+7z^{3}a^{-7}-za^{-7}+2z^{8}a^{-6}-8z^{6}a^{-6}+9z^{4}a^{-6}-5z^{2}a^{-6}+a^{-6}+z^{9}a^{-5}-3z^{7}a^{-5}+3z^{5}a^{-5}-5z^{3}a^{-5}+4za^{-5}-a^{-5}z^{-1}+3z^{8}a^{-4}-13z^{6}a^{-4}+21z^{4}a^{-4}-14z^{2}a^{-4}+3a^{-4}+z^{9}a^{-3}-5z^{7}a^{-3}+14z^{5}a^{-3}-18z^{3}a^{-3}+11za^{-3}-3a^{-3}z^{-1}+z^{8}a^{-2}-4z^{6}a^{-2}+10z^{4}a^{-2}-8z^{2}a^{-2}+3a^{-2}+3z^{5}a^{-1}+az^{3}-5z^{3}a^{-1}-az+5za^{-1}-2a^{-1}z^{-1}+2z^{4}-2z^{2}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

1

12

2

1

-1

10

2

1

8

3

2

-1

6

3

2

1

4

1

3

2

3

0

1

3

2

-2

1

-1

-4

1

Integral Khovanov Homology

(db, data source)

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