L11a280

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Link Presentations

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

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+3t(2)t(1)-t(1)-t(2)+2\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{11/2}+3q^{9/2}-5q^{7/2}+8q^{5/2}-12q^{3/2}+13{\sqrt {q}}-{\frac {14}{\sqrt {q}}}+{\frac {12}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-3a^{3}z^{3}-3z^{3}a^{-3}-2a^{3}z-za^{-3}+az^{7}+z^{7}a^{-1}+4az^{5}+4z^{5}a^{-1}+5az^{3}+4z^{3}a^{-1}+3az+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+z^{7}a^{-5}+3a^{5}z^{5}-4z^{5}a^{-5}-3a^{5}z^{3}+4z^{3}a^{-5}+a^{5}z-za^{-5}+3z^{8}a^{-4}+5a^{4}z^{6}-13z^{6}a^{-4}-5a^{4}z^{4}+16z^{4}a^{-4}+2a^{4}z^{2}-5z^{2}a^{-4}+3z^{9}a^{-3}+6a^{3}z^{7}-10z^{7}a^{-3}-7a^{3}z^{5}+7z^{5}a^{-3}+4a^{3}z^{3}+z^{3}a^{-3}-a^{3}z-za^{-3}+z^{10}a^{-2}+5a^{2}z^{8}+4z^{8}a^{-2}-4a^{2}z^{6}-24z^{6}a^{-2}-2a^{2}z^{4}+28z^{4}a^{-2}+2a^{2}z^{2}-9z^{2}a^{-2}+3az^{9}+6z^{9}a^{-1}+az^{7}-16z^{7}a^{-1}-11az^{5}+10z^{5}a^{-1}+9az^{3}-z^{3}a^{-1}-4az-2za^{-1}+az^{-1}+a^{-1}z^{-1}+z^{10}+6z^{8}-20z^{6}+16z^{4}-5z^{2}-1$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

2

-2

8

3

1

2

6

5

2

-3

4

7

3

4

2

6

5

-1

0

8

7

1

-2

6

8

2

-4

4

6

-2

-6

2

6

4

-8

1

4

-3

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$