L11a334

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{3}t(1)^{3}-3t(2)^{2}t(1)^{3}+2t(2)t(1)^{3}-4t(2)^{3}t(1)^{2}+11t(2)^{2}t(1)^{2}-11t(2)t(1)^{2}+3t(1)^{2}+3t(2)^{3}t(1)-11t(2)^{2}t(1)+11t(2)t(1)-4t(1)+2t(2)^{2}-3t(2)+1}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {8}{q^{9/2}}}-4q^{7/2}+{\frac {14}{q^{7/2}}}+9q^{5/2}-{\frac {20}{q^{5/2}}}-15q^{3/2}+{\frac {22}{q^{3/2}}}-{\frac {1}{q^{13/2}}}+{\frac {3}{q^{11/2}}}+20{\sqrt {q}}-{\frac {23}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+3az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-5a^{3}z^{3}+4az^{3}-4z^{3}a^{-1}+z^{3}a^{-3}+2a^{5}z-5a^{3}z+2az-2za^{-1}+za^{-3}+a^{5}z^{-1}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-6a^{3}z^{9}-12az^{9}-6z^{9}a^{-1}-8a^{4}z^{8}-14a^{2}z^{8}-7z^{8}a^{-2}-13z^{8}-6a^{5}z^{7}+15az^{7}+5z^{7}a^{-1}-4z^{7}a^{-3}-3a^{6}z^{6}+12a^{4}z^{6}+35a^{2}z^{6}+15z^{6}a^{-2}-z^{6}a^{-4}+36z^{6}-a^{7}z^{5}+9a^{5}z^{5}+16a^{3}z^{5}+7az^{5}+10z^{5}a^{-1}+9z^{5}a^{-3}+4a^{6}z^{4}-10a^{4}z^{4}-27a^{2}z^{4}-9z^{4}a^{-2}+2z^{4}a^{-4}-24z^{4}+2a^{7}z^{3}-8a^{5}z^{3}-21a^{3}z^{3}-13az^{3}-8z^{3}a^{-1}-6z^{3}a^{-3}-a^{6}z^{2}+2a^{4}z^{2}+6a^{2}z^{2}+2z^{2}a^{-2}-z^{2}a^{-4}+6z^{2}-a^{7}z+5a^{5}z+9a^{3}z+4az+2za^{-1}+za^{-3}+a^{4}-a^{5}z^{-1}-a^{3}z^{-1}$ (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

9

3

6

2

11

6

-5

0

12

9

3

-2

11

12

1

-4

9

11

-2

-6

5

11

6

-8

3

9

-6

-10

1

6

5

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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