L11a378

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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_16,8,17,7 X_18,6,19,5 X_22,16,11,15 X_6,18,7,17 X_8,22,9,21 X_20,10,21,9 X_10,20,1,19
Gauss code	{1, -2, 3, -4, 6, -8, 5, -9, 10, -11}, {4, -1, 2, -3, 7, -5, 8, -6, 11, -10, 9, -7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{2}t(1)^{4}-t(2)t(1)^{4}+2t(2)^{3}t(1)^{3}-4t(2)^{2}t(1)^{3}+3t(2)t(1)^{3}-t(1)^{3}+t(2)^{4}t(1)^{2}-4t(2)^{3}t(1)^{2}+5t(2)^{2}t(1)^{2}-4t(2)t(1)^{2}+t(1)^{2}-t(2)^{4}t(1)+3t(2)^{3}t(1)-4t(2)^{2}t(1)+2t(2)t(1)-t(2)^{3}+t(2)^{2}}{t(1)^{2}t(2)^{2}}}$ (db)
Jones polynomial	$4q^{9/2}-{\frac {1}{q^{9/2}}}-8q^{7/2}+{\frac {3}{q^{7/2}}}+9q^{5/2}-{\frac {6}{q^{5/2}}}-12q^{3/2}+{\frac {9}{q^{3/2}}}+q^{13/2}-2q^{11/2}+12{\sqrt {q}}-{\frac {11}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{5}-2z^{5}a^{-1}-z^{5}a^{-3}+a^{3}z^{3}-az^{3}-5z^{3}a^{-1}-2z^{3}a^{-3}+z^{3}a^{-5}+a^{3}z+az-2za^{-1}-za^{-3}+2za^{-5}+a^{-1}z^{-1}-a^{-3}z^{-1}$ (db)
Kauffman polynomial	$z^{8}a^{-6}-6z^{6}a^{-6}+12z^{4}a^{-6}-8z^{2}a^{-6}+2z^{9}a^{-5}-11z^{7}a^{-5}+21z^{5}a^{-5}+a^{5}z^{3}-17z^{3}a^{-5}+6za^{-5}+z^{10}a^{-4}-16z^{6}a^{-4}+3a^{4}z^{4}+31z^{4}a^{-4}-15z^{2}a^{-4}+6z^{9}a^{-3}-24z^{7}a^{-3}+6a^{3}z^{5}+28z^{5}a^{-3}-4a^{3}z^{3}-14z^{3}a^{-3}+a^{3}z+8za^{-3}-a^{-3}z^{-1}+z^{10}a^{-2}+7z^{8}a^{-2}+9a^{2}z^{6}-36z^{6}a^{-2}-13a^{2}z^{4}+40z^{4}a^{-2}+5a^{2}z^{2}-13z^{2}a^{-2}+a^{-2}+4z^{9}a^{-1}+10az^{7}-3z^{7}a^{-1}-20az^{5}-19z^{5}a^{-1}+10az^{3}+18z^{3}a^{-1}-3az-2za^{-1}-a^{-1}z^{-1}+8z^{8}-17z^{6}+5z^{4}-z^{2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

5

6

7

χ

14

1

-1

12

1

10

3

1

-2

8

5

1

4

6

4

3

-1

4

8

5

3

2

5

0

6

7

-1

-2

4

6

2

-4

2

5

-3

-6

1

4

3

-8

2

-2

-10

1

Integral Khovanov Homology

(db, data source)

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