L11n244

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_10,11,1,12 X_5,15,6,14 X_9,19,10,18 X_17,3,18,2 X_7,16,8,17 X₃₈₄₉ X_15,20,16,21 X_22,13,11,14 X_19,4,20,5 X_21,7,22,6
Gauss code	{1, 5, -7, 10, -3, 11, -6, 7, -4, -2}, {2, -1, 9, 3, -8, 6, -5, 4, -10, 8, -11, -9}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$0$ (db)
Jones polynomial	$-{\frac {1}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-q^{5/2}-{\frac {1}{q^{5/2}}}+2q^{3/2}-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}-2{\sqrt {q}}$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$z^{3}a^{5}+2za^{5}-z^{5}a^{3}-5z^{3}a^{3}-6za^{3}+z^{5}a+5z^{3}a+6za+az^{-1}-z^{3}a^{-1}-2za^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{9}-az^{9}-a^{4}z^{8}-3a^{2}z^{8}-2z^{8}-a^{5}z^{7}+4a^{3}z^{7}+4az^{7}-z^{7}a^{-1}-2a^{6}z^{6}+5a^{4}z^{6}+18a^{2}z^{6}+11z^{6}-a^{7}z^{5}+5a^{5}z^{5}+5a^{3}z^{5}+4az^{5}+5z^{5}a^{-1}+8a^{6}z^{4}-2a^{4}z^{4}-24a^{2}z^{4}-14z^{4}+3a^{7}z^{3}-5a^{5}z^{3}-21a^{3}z^{3}-19az^{3}-6z^{3}a^{-1}-4a^{6}z^{2}-4a^{4}z^{2}+4a^{2}z^{2}+4z^{2}+4a^{5}z+12a^{3}z+12az+4za^{-1}+1-az^{-1}-a^{-1}z^{-1}$ (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

4

χ

6

1

4

1

-1

2

1

0

3

2

1

2

-2

1

3

1

-4

2

-6

1

2

1

0

-8

1

2

1

0

-10

1

-1

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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