L11a195

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_18,13,19,14 X_6,12,1,11 X_20,15,21,16 X_14,19,15,20 X_4,18,5,17 X_16,6,17,5 X_12,21,13,22
Gauss code	{1, -4, 2, -9, 10, -6}, {4, -1, 3, -2, 6, -11, 5, -8, 7, -10, 9, -5, 8, -7, 11, -3}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{4}-3u^{2}v^{3}+4u^{2}v^{2}-4u^{2}v+2u^{2}-3uv^{4}+8uv^{3}-9uv^{2}+8uv-3u+2v^{4}-4v^{3}+4v^{2}-3v+1}{uv^{2}}}$ (db)
Jones polynomial	$q^{9/2}-3q^{7/2}+6q^{5/2}-12q^{3/2}+15{\sqrt {q}}-{\frac {19}{\sqrt {q}}}+{\frac {19}{q^{3/2}}}-{\frac {17}{q^{5/2}}}+{\frac {13}{q^{7/2}}}-{\frac {8}{q^{9/2}}}+{\frac {4}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$az^{7}-2a^{3}z^{5}+4az^{5}-2z^{5}a^{-1}+a^{5}z^{3}-5a^{3}z^{3}+8az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}+a^{5}z-4a^{3}z+8az-6za^{-1}+2za^{-3}-a^{3}z^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-a^{7}z^{3}+4a^{6}z^{6}-6a^{6}z^{4}+2a^{6}z^{2}+7a^{5}z^{7}-11a^{5}z^{5}+5a^{5}z^{3}-a^{5}z+7a^{4}z^{8}-7a^{4}z^{6}+z^{6}a^{-4}-2a^{4}z^{4}-3z^{4}a^{-4}+3a^{4}z^{2}+2z^{2}a^{-4}-a^{4}+4a^{3}z^{9}+5a^{3}z^{7}+3z^{7}a^{-3}-19a^{3}z^{5}-9z^{5}a^{-3}+14a^{3}z^{3}+9z^{3}a^{-3}-4a^{3}z-4za^{-3}+a^{3}z^{-1}+a^{2}z^{10}+12a^{2}z^{8}+4z^{8}a^{-2}-24a^{2}z^{6}-8z^{6}a^{-2}+12a^{2}z^{4}+2z^{4}a^{-2}+3a^{2}z^{2}+2z^{2}a^{-2}-3a^{2}+7az^{9}+3z^{9}a^{-1}-4az^{7}+z^{7}a^{-1}-15az^{5}-17z^{5}a^{-1}+21az^{3}+22z^{3}a^{-1}-11az-12za^{-1}+3az^{-1}+2a^{-1}z^{-1}+z^{10}+9z^{8}-22z^{6}+13z^{4}+2z^{2}-3$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

10

1

-1

8

2

6

4

1

-3

4

8

2

6

2

8

5

-3

0

11

7

4

-2

9

0

-4

8

10

-2

-6

5

9

4

-8

3

8

-5

-10

1

5

4

-12

3

-3

-14

1

Integral Khovanov Homology

(db, data source)

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