L10a92

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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_18,14,19,13 X_14,5,15,6 X_4,17,5,18 X_16,7,17,8 X_6,15,7,16 X_8,20,1,19
Gauss code	{1, -4, 2, -7, 6, -9, 8, -10}, {4, -1, 3, -2, 5, -6, 9, -8, 7, -5, 10, -3}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{3}v^{3}-2u^{3}v^{2}+u^{3}v-2u^{2}v^{3}+5u^{2}v^{2}-5u^{2}v+u^{2}+uv^{3}-5uv^{2}+5uv-2u+v^{2}-2v+1}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+3q^{5/2}-5q^{3/2}+8{\sqrt {q}}-{\frac {11}{\sqrt {q}}}+{\frac {11}{q^{3/2}}}-{\frac {11}{q^{5/2}}}+{\frac {8}{q^{7/2}}}-{\frac {6}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-2a^{3}z^{5}-7a^{3}z^{3}-7a^{3}z-a^{3}z^{-1}+az^{7}+5az^{5}-z^{5}a^{-1}+9az^{3}-3z^{3}a^{-1}+5az-2za^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+3a^{6}z^{6}-6a^{6}z^{4}+2a^{6}z^{2}+4a^{5}z^{7}-7a^{5}z^{5}+3a^{5}z^{3}-3a^{5}z+a^{5}z^{-1}+3a^{4}z^{8}-2a^{4}z^{6}-3a^{4}z^{4}+2a^{4}z^{2}-a^{4}+a^{3}z^{9}+6a^{3}z^{7}-17a^{3}z^{5}+z^{5}a^{-3}+18a^{3}z^{3}-2z^{3}a^{-3}-8a^{3}z+a^{3}z^{-1}+6a^{2}z^{8}-11a^{2}z^{6}+3z^{6}a^{-2}+8a^{2}z^{4}-7z^{4}a^{-2}+3z^{2}a^{-2}+az^{9}+6az^{7}+4z^{7}a^{-1}-19az^{5}-9z^{5}a^{-1}+22az^{3}+7z^{3}a^{-1}-7az-3za^{-1}+3z^{8}-3z^{6}-2z^{4}+3z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

2

-2

4

3

1

2

5

2

-3

0

6

3

-2

6

0

-4

5

0

-6

3

6

3

-8

3

5

-2

-10

1

4

3

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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