L11n451

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_20,12,21,11 X_22,13,19,14 X_18,22,9,21 X_12,17,13,18 X_15,8,16,5 X_7,14,8,15 X_16,19,17,20 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -8, 7}, {9, -3, 5, -4}, {-11, 2, 3, -6, 4, 8, -7, -9, 6, -5}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)(x-1)^{2}(ux-v)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-q^{5/2}+q^{3/2}-{\sqrt {q}}-{\frac {4}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {5}{q^{11/2}}}-{\frac {3}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{7}(-z)-a^{7}z^{-1}+3a^{5}z^{3}+a^{5}z^{-3}+7a^{5}z+5a^{5}z^{-1}-2a^{3}z^{5}-9a^{3}z^{3}-3a^{3}z^{-3}-15a^{3}z-10a^{3}z^{-1}+az^{5}+7az^{3}+3az^{-3}-z^{3}a^{-1}-a^{-1}z^{-3}+12az+9az^{-1}-3za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{6}a^{8}+3z^{4}a^{8}-3z^{2}a^{8}+a^{8}-3z^{7}a^{7}+10z^{5}a^{7}-9z^{3}a^{7}+5za^{7}-2a^{7}z^{-1}-2z^{8}a^{6}+z^{6}a^{6}+15z^{4}a^{6}-15z^{2}a^{6}+6a^{6}-9z^{7}a^{5}+35z^{5}a^{5}-43z^{3}a^{5}+31za^{5}-11a^{5}z^{-1}+a^{5}z^{-3}-2z^{8}a^{4}+z^{6}a^{4}+19z^{4}a^{4}-33z^{2}a^{4}-3a^{4}z^{-2}+18a^{4}-7z^{7}a^{3}+39z^{5}a^{3}-73z^{3}a^{3}+54za^{3}-18a^{3}z^{-1}+3a^{3}z^{-3}-z^{8}a^{2}+5z^{6}a^{2}+2z^{4}a^{2}-27z^{2}a^{2}-6a^{2}z^{-2}+21a^{2}-2z^{7}a+20z^{5}a-49z^{3}a+38za-14az^{-1}+3az^{-3}-z^{8}+6z^{6}-5z^{4}-6z^{2}-3z^{-2}+9-z^{7}a^{-1}+6z^{5}a^{-1}-10z^{3}a^{-1}+10za^{-1}-5a^{-1}z^{-1}+a^{-1}z^{-3}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

0

2

1

0

6

1

5

-2

4

6

1

-4

4

2

4

-6

2

4

2

-8

5

4

1

2

-10

2

4

2

-12

1

3

-2

-14

2

-16

1

-1

Integral Khovanov Homology

(db, data source)

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