L11a330

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Link Presentations

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

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)\left(u^{2}v^{4}-u^{2}v^{3}+u^{2}v^{2}-uv^{4}+3uv^{3}-3uv^{2}+3uv-u+v^{2}-v+1\right)}{u^{3/2}v^{5/2}}}$ (db)
Jones polynomial	$19q^{9/2}-22q^{7/2}+21q^{5/2}-{\frac {1}{q^{5/2}}}-19q^{3/2}+{\frac {4}{q^{3/2}}}+q^{17/2}-4q^{15/2}+9q^{13/2}-14q^{11/2}+13{\sqrt {q}}-{\frac {9}{\sqrt {q}}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z^{7}a^{-5}+4z^{5}a^{-5}+5z^{3}a^{-5}+3za^{-5}+a^{-5}z^{-1}-z^{9}a^{-3}-6z^{7}a^{-3}-13z^{5}a^{-3}-13z^{3}a^{-3}-7za^{-3}-3a^{-3}z^{-1}+z^{7}a^{-1}+4z^{5}a^{-1}+5z^{3}a^{-1}+4za^{-1}+2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$z^{4}a^{-10}+4z^{5}a^{-9}-z^{3}a^{-9}+9z^{6}a^{-8}-8z^{4}a^{-8}+3z^{2}a^{-8}+13z^{7}a^{-7}-17z^{5}a^{-7}+8z^{3}a^{-7}-za^{-7}+13z^{8}a^{-6}-18z^{6}a^{-6}+6z^{4}a^{-6}-3z^{2}a^{-6}+a^{-6}+9z^{9}a^{-5}-8z^{7}a^{-5}-9z^{5}a^{-5}+3z^{3}a^{-5}+2za^{-5}-a^{-5}z^{-1}+3z^{10}a^{-4}+12z^{8}a^{-4}-45z^{6}a^{-4}+36z^{4}a^{-4}-13z^{2}a^{-4}+3a^{-4}+15z^{9}a^{-3}-40z^{7}a^{-3}+28z^{5}a^{-3}-10z^{3}a^{-3}+7za^{-3}-3a^{-3}z^{-1}+3z^{10}a^{-2}+3z^{8}a^{-2}-31z^{6}a^{-2}+33z^{4}a^{-2}-10z^{2}a^{-2}+3a^{-2}+6z^{9}a^{-1}+az^{7}-18z^{7}a^{-1}-3az^{5}+13z^{5}a^{-1}+3az^{3}-z^{3}a^{-1}-az+3za^{-1}-2a^{-1}z^{-1}+4z^{8}-13z^{6}+12z^{4}-3z^{2}$ (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

18

1

-1

16

3

14

6

1

-5

12

8

3

5

10

11

6

-5

8

11

8

3

6

10

11

1

4

9

11

-2

2

6

12

6

0

3

7

-4

-2

1

6

5

-4

3

-3

-6

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=2$	$i=4$
$r=-4$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=-1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=0$	${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{9}$
$r=1$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=3$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=4$	${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=5$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$