L11a375

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{4}t(1)^{4}-t(2)^{3}t(1)^{4}-2t(2)^{4}t(1)^{3}+6t(2)^{3}t(1)^{3}-5t(2)^{2}t(1)^{3}+t(2)t(1)^{3}+t(2)^{4}t(1)^{2}-6t(2)^{3}t(1)^{2}+11t(2)^{2}t(1)^{2}-6t(2)t(1)^{2}+t(1)^{2}+t(2)^{3}t(1)-5t(2)^{2}t(1)+6t(2)t(1)-2t(1)-t(2)+1}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+7q^{5/2}-12q^{3/2}+15{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {18}{q^{3/2}}}-{\frac {16}{q^{5/2}}}+{\frac {12}{q^{7/2}}}-{\frac {7}{q^{9/2}}}+{\frac {3}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{9}+a^{3}z^{7}-7az^{7}+z^{7}a^{-1}+5a^{3}z^{5}-19az^{5}+5z^{5}a^{-1}+9a^{3}z^{3}-24az^{3}+9z^{3}a^{-1}+6a^{3}z-13az+6za^{-1}+a^{3}z^{-1}-az^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-2a^{7}z^{3}+a^{7}z+3a^{6}z^{6}-5a^{6}z^{4}+2a^{6}z^{2}+5a^{5}z^{7}-7a^{5}z^{5}+2a^{5}z^{3}+6a^{4}z^{8}-9a^{4}z^{6}+z^{6}a^{-4}+7a^{4}z^{4}-3z^{4}a^{-4}-4a^{4}z^{2}+2z^{2}a^{-4}+5a^{3}z^{9}-7a^{3}z^{7}+3z^{7}a^{-3}+9a^{3}z^{5}-8z^{5}a^{-3}-10a^{3}z^{3}+5z^{3}a^{-3}+6a^{3}z-za^{-3}-a^{3}z^{-1}+2a^{2}z^{10}+5a^{2}z^{8}+5z^{8}a^{-2}-17a^{2}z^{6}-13z^{6}a^{-2}+21a^{2}z^{4}+10z^{4}a^{-2}-10a^{2}z^{2}-4z^{2}a^{-2}+a^{2}+10az^{9}+5z^{9}a^{-1}-27az^{7}-12z^{7}a^{-1}+38az^{5}+13z^{5}a^{-1}-32az^{3}-13z^{3}a^{-1}+14az+6za^{-1}-az^{-1}+2z^{10}+4z^{8}-19z^{6}+22z^{4}-10z^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

5

1

-4

4

7

2

5

2

8

5

-3

0

11

7

4

-2

8

9

1

-4

8

10

-2

-6

5

9

4

-8

2

7

-5

-10

1

5

4

-12

2

-2

-14

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-6$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-3$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{8}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=0$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{11}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$