L11n86

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,3,13,4 X_14,8,15,7 X_18,10,19,9 X_19,5,20,22 X_15,21,16,20 X_21,17,22,16 X_8,18,9,17 X_10,14,11,13 X₂₅₃₆ X_4,11,1,12
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -8, 4, -9, 11, -2, 9, -3, -6, 7, 8, -4, -5, 6, -7, 5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uv^{5}-2uv^{4}+uv^{3}-uv^{2}+2uv-2u-2v^{5}+2v^{4}-v^{3}+v^{2}-2v+1}{{\sqrt {u}}v^{5/2}}}$ (db)
Jones polynomial	$q^{13/2}-3q^{11/2}+5q^{9/2}-5q^{7/2}+6q^{5/2}-6q^{3/2}+4{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$za^{-7}-z^{5}a^{-5}-4z^{3}a^{-5}-3za^{-5}+z^{7}a^{-3}+5z^{5}a^{-3}+8z^{3}a^{-3}+7za^{-3}+2a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-9z^{3}a^{-1}+4az-11za^{-1}+3az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{9}a^{-1}-z^{9}a^{-3}-4z^{8}a^{-2}-3z^{8}a^{-4}-z^{8}-az^{7}+2z^{7}a^{-1}-3z^{7}a^{-5}+15z^{6}a^{-2}+11z^{6}a^{-4}-z^{6}a^{-6}+3z^{6}+6az^{5}+6z^{5}a^{-1}+9z^{5}a^{-3}+9z^{5}a^{-5}-11z^{4}a^{-2}-14z^{4}a^{-4}-z^{4}a^{-6}+2z^{4}-12az^{3}-15z^{3}a^{-1}-9z^{3}a^{-3}-9z^{3}a^{-5}-3z^{3}a^{-7}-4z^{2}a^{-2}+8z^{2}a^{-4}+2z^{2}a^{-6}-z^{2}a^{-8}-9z^{2}+10az+13za^{-1}+4za^{-3}+2za^{-5}+za^{-7}+5a^{-2}-a^{-6}+5-3az^{-1}-5a^{-1}z^{-1}-2a^{-3}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

χ

14

1

-1

12

2

10

3

1

-2

8

2

0

6

4

3

-1

4

2

0

2

3

5

2

0

1

0

-2

3

-4

1

0

-6

1

Integral Khovanov Homology

(db, data source)

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