L11a347

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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_22,16,11,15 X_14,22,15,21 X_6,18,7,17 X_18,8,19,7 X_16,20,17,19 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, -5, 4, -8, 6, -7, 8, -3, 5, -4}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}t(1)^{2}-t(2)t(1)^{2}+2t(1)^{2}-t(2)^{2}t(1)-t(1)+2t(2)^{2}-t(2)+2\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-12q^{9/2}+7q^{7/2}-5q^{5/2}+q^{3/2}+q^{23/2}-4q^{21/2}+8q^{19/2}-12q^{17/2}+15q^{15/2}-15q^{13/2}+15q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-5}-z^{7}a^{-7}+z^{5}a^{-3}-5z^{5}a^{-5}-3z^{5}a^{-7}+z^{5}a^{-9}+5z^{3}a^{-3}-11z^{3}a^{-5}+2z^{3}a^{-9}+8za^{-3}-14za^{-5}+6za^{-7}+4a^{-3}z^{-1}-8a^{-5}z^{-1}+5a^{-7}z^{-1}-a^{-9}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-14}+4z^{5}a^{-13}-2z^{3}a^{-13}+8z^{6}a^{-12}-8z^{4}a^{-12}+2z^{2}a^{-12}+10z^{7}a^{-11}-13z^{5}a^{-11}+4z^{3}a^{-11}+za^{-11}+8z^{8}a^{-10}-9z^{6}a^{-10}+3z^{2}a^{-10}-2a^{-10}+4z^{9}a^{-9}-5z^{5}a^{-9}-5z^{3}a^{-9}+3za^{-9}+a^{-9}z^{-1}+z^{10}a^{-8}+6z^{8}a^{-8}-10z^{6}a^{-8}-11z^{4}a^{-8}+20z^{2}a^{-8}-9a^{-8}+5z^{9}a^{-7}-10z^{7}a^{-7}+3z^{5}a^{-7}+3z^{3}a^{-7}-7za^{-7}+5a^{-7}z^{-1}+z^{10}a^{-6}-z^{8}a^{-6}+5z^{6}a^{-6}-25z^{4}a^{-6}+33z^{2}a^{-6}-14a^{-6}+z^{9}a^{-5}+z^{7}a^{-5}-15z^{5}a^{-5}+27z^{3}a^{-5}-21za^{-5}+8a^{-5}z^{-1}+z^{8}a^{-4}-2z^{6}a^{-4}-5z^{4}a^{-4}+14z^{2}a^{-4}-8a^{-4}+z^{7}a^{-3}-6z^{5}a^{-3}+13z^{3}a^{-3}-12za^{-3}+4a^{-3}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

7

8

9

χ

24

1

-1

22

3

20

5

1

-4

18

7

3

4

16

8

5

-3

14

7

0

12

8

0

10

4

7

-3

8

3

8

5

6

2

4

-2

4

1

5

4

2

0

1

Integral Khovanov Homology

(db, data source)

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