L11n423

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_16,8,17,7 X_10,4,11,3 X_2,18,3,17 X_18,9,19,10 X_20,12,21,11 X_5,14,6,15 X_15,13,16,22 X_13,6,14,1 X_4,19,5,20 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 $u$ , $v$ , $w$ , ...)	${\frac {(w-1)\left(u^{2}v-u^{2}+uv^{2}w^{2}-uvw^{2}-uvw-uv+u-v^{2}w^{2}+vw^{2}\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{4}-q^{-4}+3q^{3}+3q^{-3}-3q^{2}-3q^{-2}+5q+5q^{-1}-4$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$z^{6}-a^{2}z^{4}-z^{4}a^{-2}+4z^{4}-2a^{2}z^{2}-2z^{2}a^{-2}+3z^{2}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$a^{3}z^{7}+z^{7}a^{-3}-4a^{3}z^{5}-4z^{5}a^{-3}+3a^{3}z^{3}+3z^{3}a^{-3}+3a^{2}z^{8}+3z^{8}a^{-2}-15a^{2}z^{6}-15z^{6}a^{-2}+20a^{2}z^{4}+20z^{4}a^{-2}-7a^{2}z^{2}-7z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{2}-2a^{-2}+2az^{9}+2z^{9}a^{-1}-8az^{7}-8z^{7}a^{-1}+5az^{5}+5z^{5}a^{-1}+az^{3}+z^{3}a^{-1}+2az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+6z^{8}-30z^{6}+40z^{4}-14z^{2}+2z^{-2}-3$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

χ

9

1

-1

7

2

5

2

0

3

2

1

2

1

3

4

1

-1

2

1

-3

2

4

2

-5

1

0

-7

2

-9

1

-1

Integral Khovanov Homology

(db, data source)

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