L11n430

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_9,20,10,21 X_14,5,15,6 X_12,14,7,13 X_16,8,17,7 X_22,18,13,17 X_3,10,4,11 X_18,11,19,12 X_6,15,1,16 X_4,20,5,19 X_21,3,22,2
Gauss code	{1, 11, -7, -10, 3, -9}, {5, -1, -2, 7, 8, -4}, {4, -3, 9, -5, 6, -8, 10, 2, -11, -6}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(v-1)(w-1)\left(u^{2}w^{2}-uvw+uv-uw^{2}+uw-v\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$-q^{3}+3q^{2}-5q+8-7q^{-1}+9q^{-2}-6q^{-3}+5q^{-4}-3q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{4}+2a^{4}z^{2}+a^{4}z^{-2}+a^{4}-a^{2}z^{6}-4a^{2}z^{4}-6a^{2}z^{2}-z^{2}a^{-2}-2a^{2}z^{-2}-4a^{2}-a^{-2}+2z^{4}+5z^{2}+z^{-2}+4$ (db)
Kauffman polynomial	$a^{6}z^{6}-3a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-10a^{5}z^{5}+6a^{5}z^{3}+4a^{4}z^{8}-15a^{4}z^{6}+16a^{4}z^{4}-9a^{4}z^{2}-a^{4}z^{-2}+3a^{4}+2a^{3}z^{9}-4a^{3}z^{7}-3a^{3}z^{5}+5a^{3}z^{3}+z^{3}a^{-3}-2a^{3}z-za^{-3}+2a^{3}z^{-1}+7a^{2}z^{8}-29a^{2}z^{6}+43a^{2}z^{4}+3z^{4}a^{-2}-27a^{2}z^{2}-4z^{2}a^{-2}-2a^{2}z^{-2}+7a^{2}+2a^{-2}+2az^{9}-6az^{7}+z^{7}a^{-1}+6az^{5}-z^{5}a^{-1}+2az^{3}+4z^{3}a^{-1}-4az-3za^{-1}+2az^{-1}+3z^{8}-13z^{6}+27z^{4}-21z^{2}-z^{-2}+7$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

7

1

-1

5

2

3

1

-2

1

5

2

3

-1

4

5

1

-3

5

3

2

-5

3

6

3

-7

2

3

-1

-9

1

3

2

-11

2

-2

-13

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-1$	$i=1$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{5}$
$r=-1$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=0$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{5}$
$r=1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=3$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$