L11a381

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)^{2}t(1)^{4}-t(2)t(1)^{4}-t(2)^{4}t(1)^{3}+5t(2)^{3}t(1)^{3}-8t(2)^{2}t(1)^{3}+5t(2)t(1)^{3}-t(1)^{3}+2t(2)^{4}t(1)^{2}-9t(2)^{3}t(1)^{2}+15t(2)^{2}t(1)^{2}-9t(2)t(1)^{2}+2t(1)^{2}-t(2)^{4}t(1)+5t(2)^{3}t(1)-8t(2)^{2}t(1)+5t(2)t(1)-t(1)-t(2)^{3}+t(2)^{2}}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$-10q^{9/2}+{\frac {1}{q^{9/2}}}+17q^{7/2}-{\frac {4}{q^{7/2}}}-23q^{5/2}+{\frac {10}{q^{5/2}}}+26q^{3/2}-{\frac {17}{q^{3/2}}}-q^{13/2}+4q^{11/2}-27{\sqrt {q}}+{\frac {22}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$-z^{7}a^{-1}+2az^{5}-2z^{5}a^{-1}+2z^{5}a^{-3}-a^{3}z^{3}+3az^{3}+3z^{3}a^{-3}-z^{3}a^{-5}-a^{3}z+3za^{-1}-za^{-5}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{5}a^{-7}-z^{3}a^{-7}+4z^{6}a^{-6}-4z^{4}a^{-6}+z^{2}a^{-6}+9z^{7}a^{-5}-13z^{5}a^{-5}+9z^{3}a^{-5}-3za^{-5}+12z^{8}a^{-4}+a^{4}z^{6}-18z^{6}a^{-4}-2a^{4}z^{4}+12z^{4}a^{-4}+a^{4}z^{2}-4z^{2}a^{-4}+9z^{9}a^{-3}+4a^{3}z^{7}-3z^{7}a^{-3}-8a^{3}z^{5}-15z^{5}a^{-3}+5a^{3}z^{3}+13z^{3}a^{-3}-a^{3}z-za^{-3}-a^{-3}z^{-1}+3z^{10}a^{-2}+8a^{2}z^{8}+18z^{8}a^{-2}-17a^{2}z^{6}-46z^{6}a^{-2}+12a^{2}z^{4}+33z^{4}a^{-2}-4a^{2}z^{2}-9z^{2}a^{-2}+a^{-2}+8az^{9}+17z^{9}a^{-1}-11az^{7}-27z^{7}a^{-1}+7z^{5}a^{-1}-2z^{3}a^{-1}+2az+5za^{-1}-a^{-1}z^{-1}+3z^{10}+14z^{8}-42z^{6}+31z^{4}-9z^{2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

3

-3

10

7

1

6

8

10

3

-7

6

13

7

6

4

14

11

-3

2

13

12

1

0

10

15

5

-2

7

12

-5

-4

3

10

7

-6

1

7

-6

-8

3

-10

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	$i=2$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=-1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=0$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=1$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{12}$	${\mathbb {Z} }^{13}$
$r=3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$