L11a274

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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_14,5,15,6 X_6,9,7,10 X_18,8,19,7 X_20,16,21,15 X_22,18,9,17 X_16,22,17,21 X_8,20,1,19 X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -5, 6, -10}, {5, -1, 2, -3, 11, -4, 7, -9, 8, -6, 10, -7, 9, -8}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{3}t(2)^{5}-t(1)^{2}t(2)^{5}-t(1)^{3}t(2)^{4}+3t(1)^{2}t(2)^{4}-t(1)t(2)^{4}+t(1)^{3}t(2)^{3}-3t(1)^{2}t(2)^{3}+3t(1)t(2)^{3}-t(2)^{3}-t(1)^{3}t(2)^{2}+3t(1)^{2}t(2)^{2}-3t(1)t(2)^{2}+t(2)^{2}-t(1)^{2}t(2)+3t(1)t(2)-t(2)-t(1)+1}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{9/2}-2q^{7/2}+4q^{5/2}-6q^{3/2}+7{\sqrt {q}}-{\frac {10}{\sqrt {q}}}+{\frac {9}{q^{3/2}}}-{\frac {8}{q^{5/2}}}+{\frac {6}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-8az^{7}+z^{7}a^{-1}+6a^{3}z^{5}-24az^{5}+6z^{5}a^{-1}+12a^{3}z^{3}-33az^{3}+12z^{3}a^{-1}+9a^{3}z-20az+9za^{-1}+2a^{3}z^{-1}-3az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{2}z^{10}-z^{10}-2a^{3}z^{9}-5az^{9}-3z^{9}a^{-1}-2a^{4}z^{8}+a^{2}z^{8}-3z^{8}a^{-2}-2a^{5}z^{7}+7a^{3}z^{7}+25az^{7}+14z^{7}a^{-1}-2z^{7}a^{-3}-2a^{6}z^{6}+4a^{4}z^{6}+6a^{2}z^{6}+12z^{6}a^{-2}-z^{6}a^{-4}+13z^{6}-a^{7}z^{5}+3a^{5}z^{5}-15a^{3}z^{5}-52az^{5}-26z^{5}a^{-1}+7z^{5}a^{-3}+5a^{6}z^{4}-4a^{4}z^{4}-20a^{2}z^{4}-14z^{4}a^{-2}+4z^{4}a^{-4}-29z^{4}+3a^{7}z^{3}+a^{5}z^{3}+16a^{3}z^{3}+48az^{3}+26z^{3}a^{-1}-4z^{3}a^{-3}-2a^{6}z^{2}+3a^{4}z^{2}+14a^{2}z^{2}+8z^{2}a^{-2}-3z^{2}a^{-4}+20z^{2}-2a^{7}z-10a^{3}z-23az-11za^{-1}-3a^{2}-a^{-2}-3+2a^{3}z^{-1}+3az^{-1}+a^{-1}z^{-1}$ (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

1

6

3

1

-2

4

3

1

2

4

3

-1

0

6

3

-2

4

5

1

-4

4

5

-1

-6

2

4

2

-8

2

4

-2

-10

1

3

2

-12

1

-1

-14

1

Integral Khovanov Homology

(db, data source)

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