L11a272

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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_4,9,5,10 X_14,6,15,5 X_22,14,9,13 X_18,8,19,7 X_20,18,21,17 X_16,22,17,21 X_6,16,7,15 X_8,20,1,19
Gauss code	{1, -2, 3, -4, 5, -10, 7, -11}, {4, -1, 2, -3, 6, -5, 10, -9, 8, -7, 11, -8, 9, -6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-t(2)^{2}t(1)+5t(2)t(1)-t(1)-t(2)+2\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-10q^{9/2}+14q^{7/2}-{\frac {1}{q^{7/2}}}-17q^{5/2}+{\frac {3}{q^{5/2}}}+16q^{3/2}-{\frac {7}{q^{3/2}}}+q^{15/2}-3q^{13/2}+6q^{11/2}-15{\sqrt {q}}+{\frac {11}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}-z^{7}a^{-3}+az^{5}-4z^{5}a^{-1}-4z^{5}a^{-3}+z^{5}a^{-5}+3az^{3}-6z^{3}a^{-1}-5z^{3}a^{-3}+3z^{3}a^{-5}+3az-4za^{-1}-za^{-3}+2za^{-5}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{6}a^{-8}-3z^{4}a^{-8}+2z^{2}a^{-8}+3z^{7}a^{-7}-9z^{5}a^{-7}+7z^{3}a^{-7}-za^{-7}+4z^{8}a^{-6}-10z^{6}a^{-6}+7z^{4}a^{-6}-2z^{2}a^{-6}+3z^{9}a^{-5}-3z^{7}a^{-5}-3z^{5}a^{-5}+z^{3}a^{-5}+za^{-5}+z^{10}a^{-4}+6z^{8}a^{-4}-16z^{6}a^{-4}+12z^{4}a^{-4}-3z^{2}a^{-4}+6z^{9}a^{-3}-8z^{7}a^{-3}+a^{3}z^{5}+3z^{5}a^{-3}-2a^{3}z^{3}-z^{3}a^{-3}+a^{3}z+z^{10}a^{-2}+7z^{8}a^{-2}+3a^{2}z^{6}-13z^{6}a^{-2}-5a^{2}z^{4}+7z^{4}a^{-2}+2a^{2}z^{2}+3z^{9}a^{-1}+5az^{7}+3z^{7}a^{-1}-8az^{5}-12z^{5}a^{-1}+5az^{3}+12z^{3}a^{-1}-3az-6za^{-1}+az^{-1}+a^{-1}z^{-1}+5z^{8}-5z^{6}+z^{2}-1$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

16

1

-1

14

2

12

4

1

-3

10

6

2

4

8

4

-4

6

9

6

3

4

7

8

1

2

8

9

-1

0

5

9

4

-2

2

6

-4

1

5

4

-6

2

-2

-8

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{8}$
$r=1$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$