L11a365

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X_2,13,3,14 X_14,3,15,4 X_16,5,17,6 X_6,11,7,12 X_18,8,19,7 X_22,18,11,17 X_20,10,21,9 X_8,20,9,19 X_10,22,1,21 X_4,15,5,16
Gauss code	{1, -2, 3, -11, 4, -5, 6, -9, 8, -10}, {5, -1, 2, -3, 11, -4, 7, -6, 9, -8, 10, -7}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(t(2)^{2}t(1)^{2}-t(2)t(1)^{2}+t(2)t(1)-t(1)+1\right)\left(t(1)^{2}t(2)^{2}-t(1)t(2)^{2}+t(1)t(2)-t(2)+1\right)}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+3q^{5/2}-5q^{3/2}+6{\sqrt {q}}-{\frac {8}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {5}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+6a^{3}z^{5}+11a^{3}z^{3}+6a^{3}z+a^{3}z^{-1}-az^{9}-8az^{7}+z^{7}a^{-1}-23az^{5}+6z^{5}a^{-1}-28az^{3}+11z^{3}a^{-1}-13az+6za^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+a^{7}z+2a^{6}z^{6}-6a^{6}z^{4}+3a^{6}z^{2}+2a^{5}z^{7}-5a^{5}z^{5}+2a^{5}z^{3}+2a^{4}z^{8}-6a^{4}z^{6}+z^{6}a^{-4}+6a^{4}z^{4}-4z^{4}a^{-4}-a^{4}z^{2}+3z^{2}a^{-4}+2a^{3}z^{9}-9a^{3}z^{7}+2z^{7}a^{-3}+18a^{3}z^{5}-8z^{5}a^{-3}-14a^{3}z^{3}+7z^{3}a^{-3}+6a^{3}z-za^{-3}-a^{3}z^{-1}+a^{2}z^{10}-3a^{2}z^{8}+2z^{8}a^{-2}+a^{2}z^{6}-7z^{6}a^{-2}+9a^{2}z^{4}+5z^{4}a^{-2}-6a^{2}z^{2}-z^{2}a^{-2}+a^{2}+4az^{9}+2z^{9}a^{-1}-22az^{7}-9z^{7}a^{-1}+48az^{5}+16z^{5}a^{-1}-42az^{3}-16z^{3}a^{-1}+14az+6za^{-1}-az^{-1}+z^{10}-3z^{8}+z^{6}+6z^{4}-6z^{2}$ (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

3

4

5

χ

10

1

-1

8

1

6

2

1

-1

4

3

1

2

3

2

-1

0

5

3

2

-2

3

4

1

-4

4

0

-6

2

4

2

-8

1

3

-2

-10

1

2

1

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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