L11a329

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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_20,5,21,6 X_16,8,17,7 X_18,14,19,13 X_22,16,9,15 X_8,18,1,17 X_14,22,15,21 X_6,9,7,10 X_4,19,5,20
Gauss code	{1, -2, 3, -11, 4, -10, 5, -8}, {10, -1, 2, -3, 6, -9, 7, -5, 8, -6, 11, -4, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {\left(v^{2}-v+1\right)(uv+1)(uv-u+1)(uv-v+1)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+7q^{5/2}-11q^{3/2}+14{\sqrt {q}}-{\frac {18}{\sqrt {q}}}+{\frac {17}{q^{3/2}}}-{\frac {15}{q^{5/2}}}+{\frac {11}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{3}z^{7}+5a^{3}z^{5}+9a^{3}z^{3}+7a^{3}z+2a^{3}z^{-1}-az^{9}-7az^{7}+z^{7}a^{-1}-19az^{5}+5z^{5}a^{-1}-25az^{3}+9z^{3}a^{-1}-16az+7za^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-2a^{2}z^{10}-2z^{10}-5a^{3}z^{9}-10az^{9}-5z^{9}a^{-1}-6a^{4}z^{8}-4a^{2}z^{8}-5z^{8}a^{-2}-3z^{8}-5a^{5}z^{7}+9a^{3}z^{7}+31az^{7}+14z^{7}a^{-1}-3z^{7}a^{-3}-3a^{6}z^{6}+11a^{4}z^{6}+19a^{2}z^{6}+14z^{6}a^{-2}-z^{6}a^{-4}+20z^{6}-a^{7}z^{5}+8a^{5}z^{5}-10a^{3}z^{5}-45az^{5}-18z^{5}a^{-1}+8z^{5}a^{-3}+5a^{6}z^{4}-9a^{4}z^{4}-28a^{2}z^{4}-12z^{4}a^{-2}+3z^{4}a^{-4}-29z^{4}+2a^{7}z^{3}-3a^{5}z^{3}+7a^{3}z^{3}+34az^{3}+18z^{3}a^{-1}-4z^{3}a^{-3}-a^{6}z^{2}+4a^{4}z^{2}+14a^{2}z^{2}+7z^{2}a^{-2}-2z^{2}a^{-4}+18z^{2}-a^{7}z+2a^{5}z-6a^{3}z-17az-8za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

5

1

-4

4

6

2

4

2

8

5

-3

0

10

6

4

-2

8

9

1

-4

7

9

-2

-6

4

8

4

-8

3

7

-4

-10

1

5

4

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

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