L11a328

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}t(1)^{2}-t(2)t(1)^{2}-2t(2)^{2}t(1)+4t(2)t(1)-2t(1)-t(2)+2\right)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$-3q^{9/2}+{\frac {4}{q^{9/2}}}+6q^{7/2}-{\frac {8}{q^{7/2}}}-11q^{5/2}+{\frac {12}{q^{5/2}}}+15q^{3/2}-{\frac {16}{q^{3/2}}}+q^{11/2}-{\frac {1}{q^{11/2}}}-18{\sqrt {q}}+{\frac {17}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{3}z^{5}+z^{5}a^{-3}+2a^{3}z^{3}+3z^{3}a^{-3}+2za^{-3}-az^{7}-z^{7}a^{-1}-3az^{5}-4z^{5}a^{-1}-az^{3}-6z^{3}a^{-1}+2az-4za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-6}-z^{2}a^{-6}+a^{5}z^{7}-3a^{5}z^{5}+3z^{5}a^{-5}+3a^{5}z^{3}-3z^{3}a^{-5}-a^{5}z+za^{-5}+4a^{4}z^{8}-14a^{4}z^{6}+5z^{6}a^{-4}+14a^{4}z^{4}-4z^{4}a^{-4}-4a^{4}z^{2}+z^{2}a^{-4}+5a^{3}z^{9}-14a^{3}z^{7}+7z^{7}a^{-3}+5a^{3}z^{5}-8z^{5}a^{-3}+6a^{3}z^{3}+4z^{3}a^{-3}-3a^{3}z-za^{-3}+2a^{2}z^{10}+6a^{2}z^{8}+8z^{8}a^{-2}-37a^{2}z^{6}-13z^{6}a^{-2}+38a^{2}z^{4}+8z^{4}a^{-2}-10a^{2}z^{2}-z^{2}a^{-2}+11az^{9}+6z^{9}a^{-1}-28az^{7}-6z^{7}a^{-1}+9az^{5}-10z^{5}a^{-1}+11az^{3}+15z^{3}a^{-1}-6az-6za^{-1}+az^{-1}+a^{-1}z^{-1}+2z^{10}+10z^{8}-41z^{6}+37z^{4}-9z^{2}-1$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

2

8

4

1

-3

6

7

2

5

4

8

4

-4

2

10

7

3

0

9

10

1

-2

7

8

-1

-4

5

9

4

-6

3

7

-4

-8

1

5

4

-10

3

-3

-12

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{10}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$