L11a313

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(1)+t(2))(t(1)t(2)+1)\left(t(2)^{2}-t(2)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-q^{11/2}+3q^{9/2}-6q^{7/2}+10q^{5/2}-13q^{3/2}+14{\sqrt {q}}-{\frac {16}{\sqrt {q}}}+{\frac {13}{q^{3/2}}}-{\frac {10}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {3}{q^{9/2}}}+{\frac {1}{q^{11/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-z^{5}a^{-3}-3a^{3}z^{3}-3z^{3}a^{-3}-2a^{3}z-2za^{-3}+az^{7}+z^{7}a^{-1}+4az^{5}+4z^{5}a^{-1}+5az^{3}+5z^{3}a^{-1}+2az+2za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2z^{10}a^{-2}-2z^{10}-5az^{9}-9z^{9}a^{-1}-4z^{9}a^{-3}-7a^{2}z^{8}+z^{8}a^{-2}-3z^{8}a^{-4}-3z^{8}-7a^{3}z^{7}+7az^{7}+30z^{7}a^{-1}+15z^{7}a^{-3}-z^{7}a^{-5}-5a^{4}z^{6}+13a^{2}z^{6}+13z^{6}a^{-2}+12z^{6}a^{-4}+19z^{6}-3a^{5}z^{5}+12a^{3}z^{5}+4az^{5}-33z^{5}a^{-1}-18z^{5}a^{-3}+4z^{5}a^{-5}-a^{6}z^{4}+5a^{4}z^{4}-9a^{2}z^{4}-16z^{4}a^{-2}-14z^{4}a^{-4}-17z^{4}+3a^{5}z^{3}-11a^{3}z^{3}-11az^{3}+19z^{3}a^{-1}+12z^{3}a^{-3}-4z^{3}a^{-5}+a^{6}z^{2}-a^{4}z^{2}+7z^{2}a^{-2}+5z^{2}a^{-4}+4z^{2}+4a^{3}z+4az-4za^{-1}-4za^{-3}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

2

-2

8

4

1

3

6

2

-4

4

7

4

3

2

7

6

-1

0

9

7

2

-2

6

9

3

-4

4

7

-3

-6

2

6

4

-8

1

4

-3

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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