L11n446

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_18,12,19,11 X_22,19,17,20 X_16,21,9,22 X_20,15,21,16 X_12,18,13,17 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -10, -2, 11}, {10, -1, -3, 4}, {9, -5, 6, -8, 7, -6}, {-11, 2, 5, -9, -4, 3, 8, -7}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(w-1)(x-1)(u-x)(x-v)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}+{\frac {1}{q^{9/2}}}-3q^{7/2}-{\frac {2}{q^{7/2}}}+q^{5/2}+{\frac {2}{q^{5/2}}}-2q^{3/2}-{\frac {2}{q^{3/2}}}-q^{11/2}-2{\sqrt {q}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$-a^{-5}z^{-3}-za^{-5}-2a^{-5}z^{-1}-a^{3}z^{3}+2z^{3}a^{-3}+3a^{-3}z^{-3}-2a^{3}z+7za^{-3}-a^{3}z^{-1}+8a^{-3}z^{-1}+az^{5}-z^{5}a^{-1}+5az^{3}-6z^{3}a^{-1}+az^{-3}-3a^{-1}z^{-3}+8az-12za^{-1}+6az^{-1}-11a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{8}-z^{8}a^{-2}-z^{8}a^{-4}-z^{8}-2a^{3}z^{7}-3az^{7}-4z^{7}a^{-1}-4z^{7}a^{-3}-z^{7}a^{-5}-a^{4}z^{6}+3a^{2}z^{6}+4z^{6}a^{-2}+4z^{6}a^{-4}+4z^{6}+9a^{3}z^{5}+20az^{5}+29z^{5}a^{-1}+24z^{5}a^{-3}+6z^{5}a^{-5}+4a^{4}z^{4}+6a^{2}z^{4}+6z^{4}a^{-2}+8z^{4}-9a^{3}z^{3}-36az^{3}-57z^{3}a^{-1}-42z^{3}a^{-3}-12z^{3}a^{-5}-3a^{4}z^{2}-14a^{2}z^{2}-24z^{2}a^{-2}-9z^{2}a^{-4}-26z^{2}+6a^{3}z+27az+44za^{-1}+34za^{-3}+11za^{-5}+a^{4}+6a^{2}+21a^{-2}+9a^{-4}+18-2a^{3}z^{-1}-11az^{-1}-18a^{-1}z^{-1}-14a^{-3}z^{-1}-5a^{-5}z^{-1}-6a^{-2}z^{-2}-3a^{-4}z^{-2}-3z^{-2}+az^{-3}+3a^{-1}z^{-3}+3a^{-3}z^{-3}+a^{-5}z^{-3}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

0

8

3

1

2

6

1

2

4

2

1

2

5

2

1

4

0

2

7

2

3

-2

1

3

-4

1

2

1

0

-6

1

0

-8

1

-10

1

-1

Integral Khovanov Homology

(db, data source)

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