L10a93

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Link Presentations

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Planar diagram presentation	X_10,1,11,2 X_12,4,13,3 X_20,12,9,11 X_2,9,3,10 X_4,20,5,19 X_14,5,15,6 X_18,13,19,14 X_16,7,17,8 X_6,15,7,16 X_8,17,1,18
Gauss code	{1, -4, 2, -5, 6, -9, 8, -10}, {4, -1, 3, -2, 7, -6, 9, -8, 10, -7, 5, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{3}v^{3}-2u^{3}v^{2}+u^{3}v-2u^{2}v^{3}+5u^{2}v^{2}-3u^{2}v+u^{2}+uv^{3}-3uv^{2}+5uv-2u+v^{2}-2v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {4}{\sqrt {q}}}-{\frac {7}{q^{3/2}}}+{\frac {9}{q^{5/2}}}-{\frac {10}{q^{7/2}}}+{\frac {9}{q^{9/2}}}-{\frac {8}{q^{11/2}}}+{\frac {5}{q^{13/2}}}-{\frac {3}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-z^{3}a^{7}-2za^{7}+2z^{5}a^{5}+7z^{3}a^{5}+6za^{5}+a^{5}z^{-1}-z^{7}a^{3}-5z^{5}a^{3}-9z^{3}a^{3}-7za^{3}-a^{3}z^{-1}+z^{5}a+3z^{3}a+za$ (db)
Kauffman polynomial	$-z^{4}a^{10}+z^{2}a^{10}-3z^{5}a^{9}+4z^{3}a^{9}-za^{9}-4z^{6}a^{8}+4z^{4}a^{8}-4z^{7}a^{7}+4z^{5}a^{7}-z^{3}a^{7}-3z^{8}a^{6}+3z^{6}a^{6}-2z^{4}a^{6}+z^{2}a^{6}-z^{9}a^{5}-5z^{7}a^{5}+18z^{5}a^{5}-21z^{3}a^{5}+9za^{5}-a^{5}z^{-1}-6z^{8}a^{4}+17z^{6}a^{4}-16z^{4}a^{4}+4z^{2}a^{4}+a^{4}-z^{9}a^{3}-4z^{7}a^{3}+22z^{5}a^{3}-26z^{3}a^{3}+10za^{3}-a^{3}z^{-1}-3z^{8}a^{2}+9z^{6}a^{2}-6z^{4}a^{2}+z^{2}a^{2}-3z^{7}a+11z^{5}a-10z^{3}a+2za-z^{6}+3z^{4}-z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

0

2

1

-1

-2

5

2

3

-4

5

3

-2

-6

5

4

1

-8

4

5

1

-10

4

5

-1

-12

2

5

3

-14

1

3

-2

-16

2

-18

1

-1

Integral Khovanov Homology

(db, data source)

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