L11n419

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_3,10,4,11 X_2,18,3,17 X_9,19,10,18 X_20,12,21,11 X_5,14,6,15 X_15,13,16,22 X_13,6,14,1 X_19,5,20,4 X_12,22,7,21
Gauss code	{1, -4, -3, 10, -7, 9}, {2, -1, -5, 3, 6, -11}, {-9, 7, -8, -2, 4, 5, -10, -6, 11, 8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)t(2)t(3)^{3}-t(2)t(3)^{3}+t(1)t(2)^{2}t(3)^{2}-2t(1)t(2)t(3)^{2}+t(2)t(3)^{2}-t(1)t(3)-t(1)^{2}t(2)t(3)+2t(1)t(2)t(3)+t(1)^{2}t(2)-t(1)t(2)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{6}+3q^{5}-3q^{4}+5q^{3}-4q^{2}+4q+2q^{-1}-2$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$-2z^{4}a^{-2}-7z^{2}a^{-2}+3z^{2}a^{-4}+2z^{2}-8a^{-2}+5a^{-4}-a^{-6}+4-2a^{-2}z^{-2}+a^{-4}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$z^{3}a^{-7}-za^{-7}+3z^{4}a^{-6}-4z^{2}a^{-6}+a^{-6}+z^{7}a^{-5}-3z^{5}a^{-5}+7z^{3}a^{-5}-3za^{-5}+z^{8}a^{-4}-5z^{6}a^{-4}+16z^{4}a^{-4}-18z^{2}a^{-4}-a^{-4}z^{-2}+8a^{-4}+2z^{7}a^{-3}-6z^{5}a^{-3}+11z^{3}a^{-3}-8za^{-3}+2a^{-3}z^{-1}+z^{8}a^{-2}-5z^{6}a^{-2}+16z^{4}a^{-2}-23z^{2}a^{-2}-2a^{-2}z^{-2}+12a^{-2}+z^{7}a^{-1}-3z^{5}a^{-1}+5z^{3}a^{-1}-6za^{-1}+2a^{-1}z^{-1}+3z^{4}-9z^{2}-z^{-2}+6$ (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		\

-2

-1

0

1

2

3

4

5

χ

13

1

-1

11

2

9

2

0

7

3

1

2

5

2

3

1

3

2

0

1

3

2

-1

1

0

-3

2

Integral Khovanov Homology

(db, data source)

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