L11a229

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Link Presentations

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{6}-3u^{2}v^{5}+5u^{2}v^{4}-5u^{2}v^{3}+3u^{2}v^{2}+3uv^{5}-7uv^{4}+9uv^{3}-7uv^{2}+3uv+3v^{4}-5v^{3}+5v^{2}-3v+1}{uv^{3}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+8q^{5/2}-13q^{3/2}+17{\sqrt {q}}-{\frac {21}{\sqrt {q}}}+{\frac {20}{q^{3/2}}}-{\frac {18}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-7az^{7}+z^{7}a^{-1}+5a^{3}z^{5}-20az^{5}+5z^{5}a^{-1}+10a^{3}z^{3}-29az^{3}+10z^{3}a^{-1}+9a^{3}z-21az+9za^{-1}+3a^{3}z^{-1}-5az^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+3a^{6}z^{6}-4a^{6}z^{4}+a^{6}z^{2}+6a^{5}z^{7}-10a^{5}z^{5}+8a^{5}z^{3}-3a^{5}z+7a^{4}z^{8}-9a^{4}z^{6}+z^{6}a^{-4}+4a^{4}z^{4}-3z^{4}a^{-4}+3z^{2}a^{-4}-a^{-4}+6a^{3}z^{9}-8a^{3}z^{7}+3z^{7}a^{-3}+12a^{3}z^{5}-7z^{5}a^{-3}-17a^{3}z^{3}+4z^{3}a^{-3}+12a^{3}z-3a^{3}z^{-1}+2a^{2}z^{10}+9a^{2}z^{8}+5z^{8}a^{-2}-25a^{2}z^{6}-10z^{6}a^{-2}+28a^{2}z^{4}+4z^{4}a^{-2}-18a^{2}z^{2}+5a^{2}+11az^{9}+5z^{9}a^{-1}-25az^{7}-8z^{7}a^{-1}+35az^{5}+5z^{5}a^{-1}-41az^{3}-10z^{3}a^{-1}+25az+9za^{-1}-5az^{-1}-2a^{-1}z^{-1}+2z^{10}+7z^{8}-24z^{6}+27z^{4}-20z^{2}+5$ (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

2

6

1

-5

4

7

2

5

2

10

6

-4

0

11

7

4

-2

10

11

1

-4

8

10

-2

-6

5

10

5

-8

3

8

-5

-10

5

-12

1

3

-2

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$