L11a304

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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_16,5,17,6 X_22,20,9,19 X_20,14,21,13 X_14,22,15,21 X_18,7,19,8 X_8,9,1,10 X_4,15,5,16 X_6,17,7,18
Gauss code	{1, -2, 3, -10, 4, -11, 8, -9}, {9, -1, 2, -3, 6, -7, 10, -4, 11, -8, 5, -6, 7, -5}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(uv+1)\left(u^{2}v^{4}-2u^{2}v^{3}+2u^{2}v^{2}+uv^{3}-2uv^{2}+uv+2v^{2}-2v+1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	${\sqrt {q}}-{\frac {2}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {6}{q^{5/2}}}+{\frac {7}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {8}{q^{11/2}}}-{\frac {8}{q^{13/2}}}+{\frac {6}{q^{15/2}}}-{\frac {4}{q^{17/2}}}+{\frac {2}{q^{19/2}}}-{\frac {1}{q^{21/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{7}z^{7}+6a^{7}z^{5}+12a^{7}z^{3}+9a^{7}z+2a^{7}z^{-1}-a^{5}z^{9}-8a^{5}z^{7}-24a^{5}z^{5}-33a^{5}z^{3}-19a^{5}z-3a^{5}z^{-1}+a^{3}z^{7}+6a^{3}z^{5}+11a^{3}z^{3}+6a^{3}z+a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{13}z^{3}-a^{13}z+2a^{12}z^{4}-a^{12}z^{2}+3a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+4a^{10}z^{6}-5a^{10}z^{4}+2a^{10}z^{2}+5a^{9}z^{7}-12a^{9}z^{5}+10a^{9}z^{3}-3a^{9}z+4a^{8}z^{8}-9a^{8}z^{6}+2a^{8}z^{4}+a^{8}z^{2}+3a^{7}z^{9}-9a^{7}z^{7}+9a^{7}z^{5}-13a^{7}z^{3}+10a^{7}z-2a^{7}z^{-1}+a^{6}z^{10}+a^{6}z^{8}-16a^{6}z^{6}+23a^{6}z^{4}-13a^{6}z^{2}+3a^{6}+5a^{5}z^{9}-26a^{5}z^{7}+48a^{5}z^{5}-45a^{5}z^{3}+21a^{5}z-3a^{5}z^{-1}+a^{4}z^{10}-2a^{4}z^{8}-9a^{4}z^{6}+25a^{4}z^{4}-17a^{4}z^{2}+3a^{4}+2a^{3}z^{9}-12a^{3}z^{7}+24a^{3}z^{5}-19a^{3}z^{3}+6a^{3}z-a^{3}z^{-1}+a^{2}z^{8}-6a^{2}z^{6}+11a^{2}z^{4}-6a^{2}z^{2}+a^{2}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

2

1

-1

0

1

-2

2

1

-1

-4

4

1

3

-6

4

3

-1

-8

4

3

1

-10

4

0

-12

4

0

-14

2

4

2

-16

2

4

-2

-18

2

-20

1

2

-1

-22

1

Integral Khovanov Homology

(db, data source)

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