L11a322

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Link Presentations

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(1)t(2)^{4}+t(1)^{2}t(2)^{3}-2t(1)t(2)^{3}+t(2)^{3}-2t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-2t(2)^{2}+t(1)^{2}t(2)-2t(1)t(2)+t(2)+t(1)\right)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$4q^{9/2}-{\frac {4}{q^{9/2}}}-8q^{7/2}+{\frac {9}{q^{7/2}}}+13q^{5/2}-{\frac {15}{q^{5/2}}}-19q^{3/2}+{\frac {19}{q^{3/2}}}-q^{11/2}+{\frac {1}{q^{11/2}}}+21{\sqrt {q}}-{\frac {22}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}+z^{7}a^{-1}-a^{3}z^{5}+3az^{5}+3z^{5}a^{-1}-z^{5}a^{-3}-2a^{3}z^{3}+3az^{3}+2z^{3}a^{-1}-2z^{3}a^{-3}-a^{3}z+2az-za^{-1}+az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-3z^{10}a^{-2}-3z^{10}-9az^{9}-15z^{9}a^{-1}-6z^{9}a^{-3}-14a^{2}z^{8}-2z^{8}a^{-2}-4z^{8}a^{-4}-12z^{8}-14a^{3}z^{7}+6az^{7}+41z^{7}a^{-1}+20z^{7}a^{-3}-z^{7}a^{-5}-9a^{4}z^{6}+21a^{2}z^{6}+28z^{6}a^{-2}+14z^{6}a^{-4}+44z^{6}-4a^{5}z^{5}+20a^{3}z^{5}+17az^{5}-30z^{5}a^{-1}-20z^{5}a^{-3}+3z^{5}a^{-5}-a^{6}z^{4}+7a^{4}z^{4}-7a^{2}z^{4}-31z^{4}a^{-2}-14z^{4}a^{-4}-32z^{4}+a^{5}z^{3}-11a^{3}z^{3}-14az^{3}+6z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-5}-2a^{4}z^{2}+8z^{2}a^{-2}+3z^{2}a^{-4}+7z^{2}+2a^{3}z+4az+2za^{-1}+1-az^{-1}-a^{-1}z^{-1}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

6

χ

12

1

10

3

-3

8

5

1

4

6

8

3

-5

4

11

5

6

2

10

8

-2

0

12

11

1

-2

9

12

3

-4

6

10

-4

-6

3

9

6

-8

1

6

-5

-10

3

-12

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-5$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-2$	${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=-1$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$	${\mathbb {Z} }^{9}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{12}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=4$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=5$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=6$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$