L11n197

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_12,3,13,4 X_22,10,7,9 X_10,14,11,13 X_5,19,6,18 X_21,16,22,17 X_15,20,16,21 X_19,14,20,15 X₂₇₃₈ X_4,11,5,12 X_17,1,18,6
Gauss code	{1, -9, 2, -10, -5, 11}, {9, -1, 3, -4, 10, -2, 4, 8, -7, 6, -11, 5, -8, 7, -6, -3}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{2}-u^{2}+uv^{4}-uv^{3}-uv^{2}-uv+u-v^{4}+v^{2}}{uv^{2}}}$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}-{\frac {1}{q^{7/2}}}-q^{5/2}+2q^{3/2}-{\frac {2}{q^{3/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{11/2}}}-2{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+a^{5}z^{3}+3a^{5}z+3a^{5}z^{-1}-a^{3}z^{3}-4a^{3}z-2a^{3}z^{-1}+az^{5}+4az^{3}-z^{3}a^{-1}+3az-2za^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-a^{4}z^{8}-3a^{2}z^{8}-2z^{8}-a^{7}z^{7}+6a^{3}z^{7}+4az^{7}-z^{7}a^{-1}-a^{8}z^{6}-a^{6}z^{6}+7a^{4}z^{6}+18a^{2}z^{6}+11z^{6}+5a^{7}z^{5}+a^{5}z^{5}-9a^{3}z^{5}+5z^{5}a^{-1}+5a^{8}z^{4}+6a^{6}z^{4}-11a^{4}z^{4}-28a^{2}z^{4}-16z^{4}-6a^{7}z^{3}-a^{5}z^{3}+6a^{3}z^{3}-5az^{3}-6z^{3}a^{-1}-6a^{8}z^{2}-7a^{6}z^{2}+4a^{4}z^{2}+12a^{2}z^{2}+7z^{2}+3a^{7}z+4a^{5}z-a^{3}z+2za^{-1}+a^{8}+3a^{6}+3a^{4}-a^{7}z^{-1}-3a^{5}z^{-1}-2a^{3}z^{-1}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

1

-1

2

1

0

1

2

1

0

-2

1

2

1

0

-4

1

2

-6

1

3

1

-8

1

2

-10

1

0

-12

1

0

-14

0

-16

1

-1

Integral Khovanov Homology

(db, data source)

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