L11a333

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)t(2)+1)\left(t(1)^{2}t(2)^{4}-t(1)t(2)^{4}-3t(1)^{2}t(2)^{3}+5t(1)t(2)^{3}-t(2)^{3}+3t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+3t(2)^{2}-t(1)^{2}t(2)+5t(1)t(2)-3t(2)-t(1)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-4q^{7/2}+9q^{5/2}-14q^{3/2}+19{\sqrt {q}}-{\frac {23}{\sqrt {q}}}+{\frac {22}{q^{3/2}}}-{\frac {20}{q^{5/2}}}+{\frac {14}{q^{7/2}}}-{\frac {9}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+4a^{3}z^{5}+5a^{3}z^{3}+3a^{3}z+2a^{3}z^{-1}-az^{9}-6az^{7}+z^{7}a^{-1}-13az^{5}+4z^{5}a^{-1}-13az^{3}+5z^{3}a^{-1}-8az+3za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3a^{2}z^{10}-3z^{10}-8a^{3}z^{9}-15az^{9}-7z^{9}a^{-1}-10a^{4}z^{8}-9a^{2}z^{8}-7z^{8}a^{-2}-6z^{8}-8a^{5}z^{7}+11a^{3}z^{7}+37az^{7}+14z^{7}a^{-1}-4z^{7}a^{-3}-4a^{6}z^{6}+18a^{4}z^{6}+32a^{2}z^{6}+16z^{6}a^{-2}-z^{6}a^{-4}+27z^{6}-a^{7}z^{5}+13a^{5}z^{5}-5a^{3}z^{5}-37az^{5}-9z^{5}a^{-1}+9z^{5}a^{-3}+5a^{6}z^{4}-13a^{4}z^{4}-34a^{2}z^{4}-9z^{4}a^{-2}+2z^{4}a^{-4}-27z^{4}+a^{7}z^{3}-6a^{5}z^{3}+a^{3}z^{3}+19az^{3}+7z^{3}a^{-1}-4z^{3}a^{-3}+4a^{4}z^{2}+13a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+13z^{2}+2a^{5}z-4a^{3}z-9az-3za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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

5

χ

10

1

-1

8

3

6

1

-5

4

8

3

5

2

11

6

-5

0

12

8

4

-2

11

12

1

-4

9

11

-2

-6

5

11

6

-8

4

9

-5

-10

1

6

5

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{4}$
$r=-3$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$