L11a364

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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_4,11,5,12 X_16,6,17,5 X_22,16,11,15 X_18,8,19,7 X_20,10,21,9 X_6,18,7,17 X_8,20,9,19 X_10,22,1,21
Gauss code	{1, -2, 3, -4, 5, -9, 7, -10, 8, -11}, {4, -1, 2, -3, 6, -5, 9, -7, 10, -8, 11, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{4}t(1)^{4}-t(2)^{3}t(1)^{4}-t(2)^{4}t(1)^{3}+3t(2)^{3}t(1)^{3}-2t(2)^{2}t(1)^{3}-2t(2)^{3}t(1)^{2}+3t(2)^{2}t(1)^{2}-2t(2)t(1)^{2}-2t(2)^{2}t(1)+3t(2)t(1)-t(1)-t(2)+1}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$q^{17/2}-2q^{15/2}+3q^{13/2}-5q^{11/2}+6q^{9/2}-7q^{7/2}+6q^{5/2}-6q^{3/2}+4{\sqrt {q}}-{\frac {3}{\sqrt {q}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{5/2}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-5}+6z^{5}a^{-5}+11z^{3}a^{-5}+6za^{-5}-z^{9}a^{-3}-8z^{7}a^{-3}-23z^{5}a^{-3}-28z^{3}a^{-3}-12za^{-3}-a^{-3}z^{-1}+z^{7}a^{-1}+6z^{5}a^{-1}+11z^{3}a^{-1}+7za^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-2z^{9}a^{-1}-4z^{9}a^{-3}-2z^{9}a^{-5}+3z^{8}a^{-2}+3z^{8}a^{-4}-2z^{8}a^{-6}-2z^{8}-az^{7}+10z^{7}a^{-1}+22z^{7}a^{-3}+9z^{7}a^{-5}-2z^{7}a^{-7}+z^{6}a^{-2}-z^{6}a^{-4}+6z^{6}a^{-6}-2z^{6}a^{-8}+10z^{6}+5az^{5}-16z^{5}a^{-1}-45z^{5}a^{-3}-18z^{5}a^{-5}+4z^{5}a^{-7}-2z^{5}a^{-9}-6z^{4}a^{-2}-5z^{4}a^{-4}-8z^{4}a^{-6}+3z^{4}a^{-8}-z^{4}a^{-10}-13z^{4}-6az^{3}+14z^{3}a^{-1}+43z^{3}a^{-3}+15z^{3}a^{-5}-4z^{3}a^{-7}+4z^{3}a^{-9}+5z^{2}a^{-2}+5z^{2}a^{-4}+z^{2}a^{-6}-z^{2}a^{-8}+2z^{2}a^{-10}+4z^{2}+az-7za^{-1}-14za^{-3}-5za^{-5}-za^{-9}-a^{-2}+a^{-1}z^{-1}+a^{-3}z^{-1}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

18

1

-1

16

1

14

2

1

-1

12

3

1

2

10

3

2

-1

8

4

3

1

6

3

4

1

4

3

0

2

4

2

0

1

2

-1

-2

1

2

1

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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