L11n143

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Planar diagram presentation	X₈₁₉₂ X_11,19,12,18 X_3,10,4,11 X_17,3,18,2 X_12,5,13,6 X₆₇₁₈ X_16,10,17,9 X_20,16,21,15 X_22,14,7,13 X_14,22,15,21 X_4,20,5,19
Gauss code	{1, 4, -3, -11, 5, -6}, {6, -1, 7, 3, -2, -5, 9, -10, 8, -7, -4, 2, 11, -8, 10, -9}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2u^{2}v^{2}-2u^{2}v+uv-2v+2}{uv}}$ (db)
Jones polynomial	$q^{13/2}-q^{11/2}+2q^{9/2}-3q^{7/2}+2q^{5/2}-3q^{3/2}+2{\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$z^{3}a^{-5}+3za^{-5}+a^{-5}z^{-1}-z^{5}a^{-3}-4z^{3}a^{-3}-4za^{-3}-2a^{-3}z^{-1}-z^{5}a^{-1}+az^{3}-4z^{3}a^{-1}+3az-3za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$z^{8}a^{-6}-7z^{6}a^{-6}+16z^{4}a^{-6}-13z^{2}a^{-6}+2a^{-6}+z^{9}a^{-5}-6z^{7}a^{-5}+11z^{5}a^{-5}-8z^{3}a^{-5}+4za^{-5}-a^{-5}z^{-1}+3z^{8}a^{-4}-18z^{6}a^{-4}+33z^{4}a^{-4}-22z^{2}a^{-4}+5a^{-4}+z^{9}a^{-3}-4z^{7}a^{-3}+2z^{5}a^{-3}+a^{3}z+5za^{-3}-2a^{-3}z^{-1}+2z^{8}a^{-2}-10z^{6}a^{-2}+14z^{4}a^{-2}+a^{2}z^{2}-9z^{2}a^{-2}+3a^{-2}+2z^{7}a^{-1}-9z^{5}a^{-1}+2az^{3}+10z^{3}a^{-1}-4az-4za^{-1}+az^{-1}+z^{6}-3z^{4}+z^{2}-1$ (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		\

-2

-1

0

1

2

3

4

5

6

7

χ

14

1

-1

12

0

10

2

1

-1

8

1

6

1

2

1

4

2

1

2

1

0

2

0

-2

1

-4

1

0

-6

1

Integral Khovanov Homology

(db, data source)

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