L11a287

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(uv+1)\left(uv^{2}-uv+u+v-1\right)\left(uv^{2}-uv-v^{2}+v-1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$14q^{9/2}-16q^{7/2}+15q^{5/2}-{\frac {1}{q^{5/2}}}-14q^{3/2}+{\frac {3}{q^{3/2}}}+q^{17/2}-3q^{15/2}+6q^{13/2}-11q^{11/2}+10{\sqrt {q}}-{\frac {6}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-5}+5z^{5}a^{-5}+8z^{3}a^{-5}+3za^{-5}-a^{-5}z^{-1}-z^{9}a^{-3}-7z^{7}a^{-3}-18z^{5}a^{-3}-19z^{3}a^{-3}-5za^{-3}+a^{-3}z^{-1}+z^{7}a^{-1}+5z^{5}a^{-1}+8z^{3}a^{-1}+4za^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}a^{-4}-4z^{9}a^{-1}-9z^{9}a^{-3}-5z^{9}a^{-5}+z^{8}a^{-2}-3z^{8}a^{-4}-7z^{8}a^{-6}-3z^{8}-az^{7}+15z^{7}a^{-1}+29z^{7}a^{-3}+6z^{7}a^{-5}-7z^{7}a^{-7}+11z^{6}a^{-2}+14z^{6}a^{-4}+10z^{6}a^{-6}-5z^{6}a^{-8}+12z^{6}+4az^{5}-19z^{5}a^{-1}-35z^{5}a^{-3}+9z^{5}a^{-7}-3z^{5}a^{-9}-12z^{4}a^{-2}-7z^{4}a^{-4}-4z^{4}a^{-6}+4z^{4}a^{-8}-z^{4}a^{-10}-14z^{4}-4az^{3}+13z^{3}a^{-1}+26z^{3}a^{-3}+z^{3}a^{-5}-5z^{3}a^{-7}+3z^{3}a^{-9}+4z^{2}a^{-2}-2z^{2}a^{-6}-z^{2}a^{-8}+z^{2}a^{-10}+4z^{2}-5za^{-1}-5za^{-3}-za^{-7}-za^{-9}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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

4

1

-3

12

7

2

5

10

8

5

-3

8

6

2

6

7

8

1

4

7

8

-1

2

4

8

4

0

2

6

-4

-2

1

4

3

-4

2

-2

-6

1

Integral Khovanov Homology

(db, data source)

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