L10a59

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_14,6,15,5 X_16,13,17,14 X_20,18,7,17 X_18,12,19,11 X_12,20,13,19 X_4,16,5,15 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -9, 2, -8, 3, -10}, {9, -1, 10, -2, 6, -7, 4, -3, 8, -4, 5, -6, 7, -5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{3}-3u^{2}v^{2}+3u^{2}v-u^{2}+uv^{4}-5uv^{3}+7uv^{2}-5uv+u-v^{4}+3v^{3}-3v^{2}+v}{uv^{2}}}$ (db)
Jones polynomial	$q^{11/2}-3q^{9/2}+6q^{7/2}-9q^{5/2}+11q^{3/2}-12{\sqrt {q}}+{\frac {10}{\sqrt {q}}}-{\frac {9}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {3}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$az^{5}+z^{5}a^{-1}-a^{3}z^{3}+2az^{3}-2z^{3}a^{-3}-a^{3}z+3az-3za^{-1}-za^{-3}+za^{-5}+2az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-3a^{2}z^{8}-3z^{8}a^{-2}-6z^{8}-3a^{3}z^{7}-6az^{7}-8z^{7}a^{-1}-5z^{7}a^{-3}-a^{4}z^{6}+6a^{2}z^{6}-4z^{6}a^{-2}-5z^{6}a^{-4}+8z^{6}+10a^{3}z^{5}+23az^{5}+20z^{5}a^{-1}+4z^{5}a^{-3}-3z^{5}a^{-5}+3a^{4}z^{4}+a^{2}z^{4}+15z^{4}a^{-2}+6z^{4}a^{-4}-z^{4}a^{-6}+6z^{4}-10a^{3}z^{3}-23az^{3}-17z^{3}a^{-1}-z^{3}a^{-3}+3z^{3}a^{-5}-2a^{4}z^{2}-4a^{2}z^{2}-12z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-9z^{2}+3a^{3}z+11az+10za^{-1}+za^{-3}-za^{-5}+3a^{-2}+a^{-4}+3-2az^{-1}-3a^{-1}z^{-1}-a^{-3}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

4

1

-3

6

5

2

3

4

6

4

-2

2

6

5

1

0

5

7

2

-2

4

5

-1

-4

2

6

4

-6

1

3

-2

-8

2

-10

1

-1

Integral Khovanov Homology

(db, data source)

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