L11a346

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Link Presentations

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Planar diagram presentation	X_12,1,13,2 X₈₄₉₃ X_20,14,21,13 X_22,17,11,18 X_18,21,19,22 X_6,16,7,15 X_16,8,17,7 X_14,20,15,19 X_10,6,1,5 X_4,10,5,9 X_2,11,3,12
Gauss code	{1, -11, 2, -10, 9, -6, 7, -2, 10, -9}, {11, -1, 3, -8, 6, -7, 4, -5, 8, -3, 5, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}t(1)^{2}-4t(2)t(1)^{2}+t(1)^{2}+t(2)t(1)+t(2)^{2}-4t(2)+1\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$15q^{9/2}-17q^{7/2}+16q^{5/2}-{\frac {1}{q^{5/2}}}-14q^{3/2}+{\frac {2}{q^{3/2}}}+q^{17/2}-3q^{15/2}+7q^{13/2}-12q^{11/2}+10{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{3}a^{-7}+2za^{-7}+a^{-7}z^{-1}-2z^{5}a^{-5}-6z^{3}a^{-5}-7za^{-5}-4a^{-5}z^{-1}+z^{7}a^{-3}+4z^{5}a^{-3}+8z^{3}a^{-3}+11za^{-3}+6a^{-3}z^{-1}-2z^{5}a^{-1}+az^{3}-7z^{3}a^{-1}+3az-9za^{-1}+2az^{-1}-5a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-2z^{9}a^{-1}-6z^{9}a^{-3}-4z^{9}a^{-5}-5z^{8}a^{-2}-11z^{8}a^{-4}-8z^{8}a^{-6}-2z^{8}-az^{7}+z^{7}a^{-1}+4z^{7}a^{-3}-7z^{7}a^{-5}-9z^{7}a^{-7}+20z^{6}a^{-2}+28z^{6}a^{-4}+9z^{6}a^{-6}-6z^{6}a^{-8}+7z^{6}+5az^{5}+17z^{5}a^{-1}+31z^{5}a^{-3}+37z^{5}a^{-5}+15z^{5}a^{-7}-3z^{5}a^{-9}-13z^{4}a^{-2}-10z^{4}a^{-4}+3z^{4}a^{-6}+6z^{4}a^{-8}-z^{4}a^{-10}-7z^{4}-9az^{3}-33z^{3}a^{-1}-48z^{3}a^{-3}-41z^{3}a^{-5}-15z^{3}a^{-7}+2z^{3}a^{-9}-z^{2}a^{-2}-6z^{2}a^{-4}-9z^{2}a^{-6}-4z^{2}a^{-8}+z^{2}a^{-10}+z^{2}+7az+22za^{-1}+29za^{-3}+20za^{-5}+6za^{-7}+a^{-2}+3a^{-4}+3a^{-6}+a^{-8}+1-2az^{-1}-5a^{-1}z^{-1}-6a^{-3}z^{-1}-4a^{-5}z^{-1}-a^{-7}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

2

14

5

1

-4

12

7

2

5

10

8

5

-3

8

9

7

2

6

7

8

1

4

7

9

-2

2

5

9

4

0

1

5

-4

-2

1

5

4

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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