L11a200

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_16,5,17,6 X_22,18,7,17 X_18,12,19,11 X_20,14,21,13 X_12,20,13,19 X_14,22,15,21 X₆₇₁₈ X_4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 6, -8, 7, -9, 11, -4, 5, -6, 8, -7, 9, -5}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{6}-2u^{2}v^{5}+2u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}+2uv^{5}-3uv^{4}+3uv^{3}-3uv^{2}+2uv+2v^{4}-2v^{3}+2v^{2}-2v+1}{uv^{3}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {4}{q^{9/2}}}-2q^{7/2}+{\frac {6}{q^{7/2}}}+4q^{5/2}-{\frac {9}{q^{5/2}}}-6q^{3/2}+{\frac {9}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}+8{\sqrt {q}}-{\frac {10}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-8az^{7}+z^{7}a^{-1}+6a^{3}z^{5}-24az^{5}+6z^{5}a^{-1}+12a^{3}z^{3}-33az^{3}+12z^{3}a^{-1}+9a^{3}z-21az+9za^{-1}+3a^{3}z^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+a^{7}z+2a^{6}z^{6}-5a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-9a^{5}z^{5}+8a^{5}z^{3}-3a^{5}z+3a^{4}z^{8}-9a^{4}z^{6}+z^{6}a^{-4}+9a^{4}z^{4}-4z^{4}a^{-4}-a^{4}z^{2}+4z^{2}a^{-4}-a^{-4}+3a^{3}z^{9}-13a^{3}z^{7}+2z^{7}a^{-3}+26a^{3}z^{5}-7z^{5}a^{-3}-22a^{3}z^{3}+5z^{3}a^{-3}+12a^{3}z-3a^{3}z^{-1}+a^{2}z^{10}+2z^{8}a^{-2}-11a^{2}z^{6}-5z^{6}a^{-2}+29a^{2}z^{4}-20a^{2}z^{2}+2z^{2}a^{-2}+5a^{2}+5az^{9}+2z^{9}a^{-1}-25az^{7}-7z^{7}a^{-1}+55az^{5}+12z^{5}a^{-1}-54az^{3}-16z^{3}a^{-1}+25az+9za^{-1}-5az^{-1}-2a^{-1}z^{-1}+z^{10}-z^{8}-6z^{6}+19z^{4}-20z^{2}+5$ (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

5

χ

10

1

-1

8

1

6

3

1

-2

4

3

1

2

5

3

-2

0

5

3

2

-2

5

6

1

-4

4

0

-6

2

5

3

-8

2

4

-2

-10

2

-12

1

2

-1

-14

1

Integral Khovanov Homology

(db, data source)

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