L11a308

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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_6,9,7,10 X_20,7,21,8 X_8,19,1,20 X_18,13,19,14 X_16,6,17,5 X_4,18,5,17 X_22,15,9,16 X_14,21,15,22
Gauss code	{1, -2, 3, -9, 8, -4, 5, -6}, {4, -1, 2, -3, 7, -11, 10, -8, 9, -7, 6, -5, 11, -10}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)+t(2)-1)(t(1)t(2)+1)(t(2)t(1)-t(1)-t(2))\left(t(2)^{2}-t(2)+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {1}{\sqrt {q}}}+{\frac {3}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {10}{q^{7/2}}}-{\frac {15}{q^{9/2}}}+{\frac {17}{q^{11/2}}}-{\frac {17}{q^{13/2}}}+{\frac {15}{q^{15/2}}}-{\frac {12}{q^{17/2}}}+{\frac {7}{q^{19/2}}}-{\frac {3}{q^{21/2}}}+{\frac {1}{q^{23/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}\left(-z^{5}\right)-3a^{9}z^{3}-3a^{9}z-a^{9}z^{-1}+a^{7}z^{7}+4a^{7}z^{5}+7a^{7}z^{3}+8a^{7}z+3a^{7}z^{-1}+a^{5}z^{7}+3a^{5}z^{5}-5a^{5}z-2a^{5}z^{-1}-a^{3}z^{5}-3a^{3}z^{3}-2a^{3}z$ (db)
Kauffman polynomial	$a^{14}z^{4}-a^{14}z^{2}+3a^{13}z^{5}-2a^{13}z^{3}+6a^{12}z^{6}-6a^{12}z^{4}+3a^{12}z^{2}+9a^{11}z^{7}-15a^{11}z^{5}+13a^{11}z^{3}-3a^{11}z+9a^{10}z^{8}-16a^{10}z^{6}+12a^{10}z^{4}-4a^{10}z^{2}+a^{10}+6a^{9}z^{9}-7a^{9}z^{7}-3a^{9}z^{5}+4a^{9}z^{3}-a^{9}z^{-1}+2a^{8}z^{10}+7a^{8}z^{8}-29a^{8}z^{6}+29a^{8}z^{4}-17a^{8}z^{2}+3a^{8}+10a^{7}z^{9}-30a^{7}z^{7}+31a^{7}z^{5}-23a^{7}z^{3}+12a^{7}z-3a^{7}z^{-1}+2a^{6}z^{10}+a^{6}z^{8}-18a^{6}z^{6}+21a^{6}z^{4}-11a^{6}z^{2}+3a^{6}+4a^{5}z^{9}-13a^{5}z^{7}+12a^{5}z^{5}-7a^{5}z^{3}+7a^{5}z-2a^{5}z^{-1}+3a^{4}z^{8}-11a^{4}z^{6}+11a^{4}z^{4}-2a^{4}z^{2}+a^{3}z^{7}-4a^{3}z^{5}+5a^{3}z^{3}-2a^{3}z$ (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		\

-9

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

2

-2

-4

5

1

4

-6

6

3

-3

-8

9

4

5

-10

8

6

-2

-12

9

0

-14

7

9

2

-16

5

8

-3

-18

2

7

5

-20

1

5

-4

-22

2

-24

1

-1

Integral Khovanov Homology

(db, data source)

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