L11a283

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Link Presentations

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{5}-t(1)t(2)^{5}+t(1)^{3}t(2)^{4}-4t(1)^{2}t(2)^{4}+5t(1)t(2)^{4}-t(2)^{4}-2t(1)^{3}t(2)^{3}+7t(1)^{2}t(2)^{3}-7t(1)t(2)^{3}+2t(2)^{3}+2t(1)^{3}t(2)^{2}-7t(1)^{2}t(2)^{2}+7t(1)t(2)^{2}-2t(2)^{2}-t(1)^{3}t(2)+5t(1)^{2}t(2)-4t(1)t(2)+t(2)-t(1)^{2}+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$3q^{9/2}-{\frac {4}{q^{9/2}}}-7q^{7/2}+{\frac {9}{q^{7/2}}}+11q^{5/2}-{\frac {14}{q^{5/2}}}-17q^{3/2}+{\frac {18}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+19{\sqrt {q}}-{\frac {20}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+4z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+2az^{3}+7z^{3}a^{-1}-3z^{3}a^{-3}-a^{3}z-az+7za^{-1}-3za^{-3}-az^{-1}+3a^{-1}z^{-1}-2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}+z^{7}a^{-5}+4a^{5}z^{5}-4z^{5}a^{-5}-a^{5}z^{3}+5z^{3}a^{-5}-2za^{-5}+3z^{8}a^{-4}+9a^{4}z^{6}-11z^{6}a^{-4}-8a^{4}z^{4}+12z^{4}a^{-4}+3a^{4}z^{2}-3z^{2}a^{-4}+4z^{9}a^{-3}+13a^{3}z^{7}-12z^{7}a^{-3}-18a^{3}z^{5}+10z^{5}a^{-3}+9a^{3}z^{3}-4z^{3}a^{-3}-a^{3}z+5za^{-3}-2a^{-3}z^{-1}+2z^{10}a^{-2}+12a^{2}z^{8}+3z^{8}a^{-2}-16a^{2}z^{6}-23z^{6}a^{-2}+4a^{2}z^{4}+26z^{4}a^{-2}-a^{2}z^{2}-11z^{2}a^{-2}+a^{2}+3a^{-2}+7az^{9}+11z^{9}a^{-1}-2az^{7}-28z^{7}a^{-1}-14az^{5}+22z^{5}a^{-1}+4az^{3}-15z^{3}a^{-1}+2az+10za^{-1}-az^{-1}-3a^{-1}z^{-1}+2z^{10}+12z^{8}-37z^{6}+27z^{4}-12z^{2}+3$ (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

12

1

10

2

-2

8

5

1

4

6

7

3

-4

4

10

4

6

2

9

7

-2

0

11

10

1

-2

8

10

2

-4

6

10

-4

-6

3

8

5

-8

1

6

-5

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

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