L11a309

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}-3t(1)t(2)^{3}+t(2)^{3}-t(1)^{2}t(2)^{2}+5t(1)t(2)^{2}-t(2)^{2}+t(1)^{2}t(2)-3t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{7/2}+4q^{5/2}-9q^{3/2}+15{\sqrt {q}}-{\frac {22}{\sqrt {q}}}+{\frac {24}{q^{3/2}}}-{\frac {25}{q^{5/2}}}+{\frac {21}{q^{7/2}}}-{\frac {16}{q^{9/2}}}+{\frac {10}{q^{11/2}}}-{\frac {4}{q^{13/2}}}+{\frac {1}{q^{15/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{5}z^{5}-2a^{5}z^{3}-2a^{5}z-a^{5}z^{-1}+a^{3}z^{7}+3a^{3}z^{5}+4a^{3}z^{3}+4a^{3}z+3a^{3}z^{-1}+az^{7}+3az^{5}-z^{5}a^{-1}+3az^{3}-2z^{3}a^{-1}-az-za^{-1}-2az^{-1}$ (db)
Kauffman polynomial	$a^{8}z^{6}-2a^{8}z^{4}+a^{8}z^{2}+4a^{7}z^{7}-8a^{7}z^{5}+4a^{7}z^{3}+8a^{6}z^{8}-18a^{6}z^{6}+13a^{6}z^{4}-6a^{6}z^{2}+a^{6}+8a^{5}z^{9}-14a^{5}z^{7}+7a^{5}z^{5}-7a^{5}z^{3}+4a^{5}z-a^{5}z^{-1}+3a^{4}z^{10}+11a^{4}z^{8}-37a^{4}z^{6}+31a^{4}z^{4}-13a^{4}z^{2}+3a^{4}+16a^{3}z^{9}-31a^{3}z^{7}+20a^{3}z^{5}+z^{5}a^{-3}-10a^{3}z^{3}-z^{3}a^{-3}+7a^{3}z-3a^{3}z^{-1}+3a^{2}z^{10}+13a^{2}z^{8}-36a^{2}z^{6}+4z^{6}a^{-2}+28a^{2}z^{4}-5z^{4}a^{-2}-8a^{2}z^{2}+z^{2}a^{-2}+3a^{2}+8az^{9}-5az^{7}+8z^{7}a^{-1}-8az^{5}-12z^{5}a^{-1}+9az^{3}+7z^{3}a^{-1}+az-2za^{-1}-2az^{-1}+10z^{8}-14z^{6}+7z^{4}-z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

8

1

6

3

-3

4

6

1

5

2

9

3

-6

0

13

6

7

-2

13

11

-2

-4

12

11

1

-6

9

13

4

-8

7

12

-5

-10

3

9

6

-12

1

7

-6

-14

3

-16

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-4$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-3$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=-2$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{12}$
$r=-1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{13}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$