L10a86

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,6,17,5 X_18,10,19,9 X_14,19,15,20 X_10,16,11,15 X_20,11,7,12 X₄₇₅₈ X_2,14,3,13 X_12,4,13,3 X_6,18,1,17
Gauss code	{1, -8, 9, -7, 2, -10}, {7, -1, 3, -5, 6, -9, 8, -4, 5, -2, 10, -3, 4, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{4}-3u^{2}v^{3}+4u^{2}v^{2}-3u^{2}v+u^{2}-2uv^{4}+7uv^{3}-11uv^{2}+7uv-2u+v^{4}-3v^{3}+4v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	$-9q^{9/2}+13q^{7/2}-{\frac {1}{q^{7/2}}}-17q^{5/2}+{\frac {4}{q^{5/2}}}+18q^{3/2}-{\frac {9}{q^{3/2}}}-q^{13/2}+4q^{11/2}-17{\sqrt {q}}+{\frac {13}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+az^{5}-4z^{5}a^{-1}+2z^{5}a^{-3}+2az^{3}-7z^{3}a^{-1}+5z^{3}a^{-3}-z^{3}a^{-5}+2az-5za^{-1}+5za^{-3}-za^{-5}+az^{-1}-2a^{-1}z^{-1}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-3z^{9}a^{-1}-3z^{9}a^{-3}-16z^{8}a^{-2}-8z^{8}a^{-4}-8z^{8}-8az^{7}-12z^{7}a^{-1}-12z^{7}a^{-3}-8z^{7}a^{-5}-4a^{2}z^{6}+30z^{6}a^{-2}+11z^{6}a^{-4}-4z^{6}a^{-6}+11z^{6}-a^{3}z^{5}+14az^{5}+36z^{5}a^{-1}+36z^{5}a^{-3}+14z^{5}a^{-5}-z^{5}a^{-7}+5a^{2}z^{4}-18z^{4}a^{-2}-4z^{4}a^{-4}+5z^{4}a^{-6}-4z^{4}+a^{3}z^{3}-9az^{3}-31z^{3}a^{-1}-31z^{3}a^{-3}-9z^{3}a^{-5}+z^{3}a^{-7}-a^{2}z^{2}+4z^{2}a^{-2}+z^{2}a^{-4}-z^{2}a^{-6}+z^{2}+4az+12za^{-1}+12za^{-3}+4za^{-5}-a^{-2}-az^{-1}-2a^{-1}z^{-1}-2a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

6

1

5

8

7

3

-4

6

10

6

4

9

8

-1

2

8

9

-1

0

6

10

4

-2

3

7

-4

1

6

5

-6

3

-3

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{10}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$