L11n129

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_4,15,5,16 X_5,13,6,12 X_16,12,17,11 X_17,6,18,1 X_19,14,20,15 X_13,20,14,21 X_21,19,22,18
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, 7, 6, -10, 9, 5, -7, -8, 11, -9, 10, -11, -3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {u^{2}v^{2}-2u^{2}v+u^{2}-3uv^{2}+7uv-3u+v^{2}-2v+1}{uv}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {1}{q^{9/2}}}-3q^{7/2}+{\frac {2}{q^{7/2}}}+4q^{5/2}-{\frac {5}{q^{5/2}}}-6q^{3/2}+{\frac {6}{q^{3/2}}}+7{\sqrt {q}}-{\frac {7}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-1}+z^{3}a^{-3}-3a^{3}z-2a^{3}z^{-1}+za^{-3}-z^{5}a^{-1}+3az^{3}-3z^{3}a^{-1}+5az+2az^{-1}-4za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-2a^{2}z^{8}-3z^{8}a^{-2}-5z^{8}-a^{3}z^{7}-2z^{7}a^{-1}-3z^{7}a^{-3}+8a^{2}z^{6}+10z^{6}a^{-2}-z^{6}a^{-4}+19z^{6}+3a^{3}z^{5}+10az^{5}+18z^{5}a^{-1}+11z^{5}a^{-3}-2a^{4}z^{4}-15a^{2}z^{4}-7z^{4}a^{-2}+3z^{4}a^{-4}-23z^{4}-a^{5}z^{3}-9a^{3}z^{3}-21az^{3}-22z^{3}a^{-1}-9z^{3}a^{-3}+2a^{4}z^{2}+8a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+9z^{2}+2a^{5}z+8a^{3}z+12az+8za^{-1}+2za^{-3}-a^{2}-a^{5}z^{-1}-2a^{3}z^{-1}-2az^{-1}-a^{-1}z^{-1}$ (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

χ

10

1

-1

8

2

6

2

1

-1

4

2

3

2

-1

0

4

0

-2

3

4

1

-4

2

3

-1

-6

1

4

3

-8

1

-1

-10

1

Integral Khovanov Homology

(db, data source)

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