L11a301

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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_4,9,5,10 X_16,6,17,5 X_20,14,21,13 X_22,16,9,15 X_14,22,15,21 X_18,8,19,7 X_6,18,7,17 X_8,20,1,19
Gauss code	{1, -2, 3, -4, 5, -10, 9, -11}, {4, -1, 2, -3, 6, -8, 7, -5, 10, -9, 11, -6, 8, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(1)t(2)+1)^{2}\left(t(2)^{2}+1\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$9q^{9/2}-10q^{7/2}+9q^{5/2}-{\frac {1}{q^{5/2}}}-9q^{3/2}+{\frac {2}{q^{3/2}}}+q^{17/2}-2q^{15/2}+4q^{13/2}-7q^{11/2}+6{\sqrt {q}}-{\frac {4}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-z^{9}a^{-3}+z^{7}a^{-1}-8z^{7}a^{-3}+z^{7}a^{-5}+6z^{5}a^{-1}-24z^{5}a^{-3}+6z^{5}a^{-5}+12z^{3}a^{-1}-32z^{3}a^{-3}+12z^{3}a^{-5}+9za^{-1}-17za^{-3}+8za^{-5}+2a^{-1}z^{-1}-3a^{-3}z^{-1}+a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{10}a^{-2}-z^{10}a^{-4}-2z^{9}a^{-1}-5z^{9}a^{-3}-3z^{9}a^{-5}+z^{8}a^{-2}-z^{8}a^{-4}-4z^{8}a^{-6}-2z^{8}-az^{7}+8z^{7}a^{-1}+23z^{7}a^{-3}+10z^{7}a^{-5}-4z^{7}a^{-7}+6z^{6}a^{-2}+12z^{6}a^{-4}+12z^{6}a^{-6}-3z^{6}a^{-8}+9z^{6}+5az^{5}-11z^{5}a^{-1}-46z^{5}a^{-3}-18z^{5}a^{-5}+10z^{5}a^{-7}-2z^{5}a^{-9}-10z^{4}a^{-2}-22z^{4}a^{-4}-17z^{4}a^{-6}+5z^{4}a^{-8}-z^{4}a^{-10}-11z^{4}-7az^{3}+12z^{3}a^{-1}+51z^{3}a^{-3}+17z^{3}a^{-5}-12z^{3}a^{-7}+3z^{3}a^{-9}+9z^{2}a^{-2}+17z^{2}a^{-4}+6z^{2}a^{-6}-3z^{2}a^{-8}+2z^{2}a^{-10}+3z^{2}+2az-9za^{-1}-21za^{-3}-8za^{-5}+2za^{-7}-3a^{-2}-3a^{-4}-a^{-6}+2a^{-1}z^{-1}+3a^{-3}z^{-1}+a^{-5}z^{-1}$ (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

3

1

-2

12

4

1

3

10

5

3

-2

8

5

4

1

6

4

5

1

4

5

0

2

3

6

3

0

1

3

-2

1

3

2

-4

1

-1

-6

1

Integral Khovanov Homology

(db, data source)

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