L11a310

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(uv+1)(uv-u-2v+1)(uv-2u-v+1)}{u^{3/2}v^{3/2}}}$ (db)
Jones polynomial	$q^{17/2}-3q^{15/2}+6q^{13/2}-11q^{11/2}+14q^{9/2}-16q^{7/2}+15q^{5/2}-14q^{3/2}+10{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-3}-2z^{5}a^{-1}+4z^{5}a^{-3}-2z^{5}a^{-5}+az^{3}-6z^{3}a^{-1}+7z^{3}a^{-3}-6z^{3}a^{-5}+z^{3}a^{-7}+2az-4za^{-1}+7za^{-3}-5za^{-5}+2za^{-7}+a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-3z^{9}a^{-1}-7z^{9}a^{-3}-4z^{9}a^{-5}-7z^{8}a^{-2}-11z^{8}a^{-4}-7z^{8}a^{-6}-3z^{8}-az^{7}+7z^{7}a^{-1}+13z^{7}a^{-3}-2z^{7}a^{-5}-7z^{7}a^{-7}+32z^{6}a^{-2}+35z^{6}a^{-4}+10z^{6}a^{-6}-5z^{6}a^{-8}+12z^{6}+4az^{5}+2z^{5}a^{-1}+7z^{5}a^{-3}+21z^{5}a^{-5}+9z^{5}a^{-7}-3z^{5}a^{-9}-33z^{4}a^{-2}-28z^{4}a^{-4}-5z^{4}a^{-6}+4z^{4}a^{-8}-z^{4}a^{-10}-15z^{4}-5az^{3}-9z^{3}a^{-1}-16z^{3}a^{-3}-21z^{3}a^{-5}-6z^{3}a^{-7}+3z^{3}a^{-9}+10z^{2}a^{-2}+6z^{2}a^{-4}-z^{2}a^{-8}+z^{2}a^{-10}+6z^{2}+2az+3za^{-1}+7za^{-3}+8za^{-5}+za^{-7}-za^{-9}+a^{-4}-a^{-3}z^{-1}-a^{-5}z^{-1}$ (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)

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