L11n52

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_12,8,13,7 X_13,18,14,19 X_16,9,17,10 X_8,17,9,18 X_19,22,20,5 X_15,21,16,20 X_21,15,22,14 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -3, -4, 9, -8, -5, 6, 4, -7, 8, -9, 7}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)^{3}}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-2q^{5/2}+3q^{3/2}-5{\sqrt {q}}+{\frac {5}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {5}{q^{5/2}}}-{\frac {4}{q^{7/2}}}+{\frac {2}{q^{9/2}}}-{\frac {1}{q^{11/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{5}+a^{5}z^{-1}-2z^{3}a^{3}-4za^{3}-2a^{3}z^{-1}+z^{5}a+3z^{3}a+3za+az^{-1}-z^{3}a^{-1}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{5}z^{7}-5a^{5}z^{5}+8a^{5}z^{3}-5a^{5}z+a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+11a^{4}z^{4}-5a^{4}z^{2}+a^{4}+a^{3}z^{9}-16a^{3}z^{5}+27a^{3}z^{3}-15a^{3}z+za^{-3}+2a^{3}z^{-1}-a^{-3}z^{-1}+5a^{2}z^{8}-22a^{2}z^{6}+26a^{2}z^{4}+z^{4}a^{-2}-12a^{2}z^{2}-z^{2}a^{-2}+3a^{2}+a^{-2}+az^{9}+az^{7}+2z^{7}a^{-1}-19az^{5}-8z^{5}a^{-1}+28az^{3}+9z^{3}a^{-1}-12az-za^{-1}+az^{-1}-a^{-1}z^{-1}+3z^{8}-13z^{6}+16z^{4}-8z^{2}+2$ (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

χ

6

2

4

1

-1

2

4

2

0

3

0

-2

2

3

1

0

-4

3

0

-6

1

2

-1

-8

1

3

2

-10

1

-1

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$	$i=2$
$r=-6$		${\mathbb {Z} }$
$r=-5$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$		${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$		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}\oplus{\mathbb Z}_2^{2}}	${\mathbb {Z} }^{2}$
$r=-1$		${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=0$	${\mathbb {Z} }$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=1$		${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=2$		${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$