L11a168

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_18,11,19,12 X_10,4,11,3 X_2,17,3,18 X_14,5,15,6 X₆₇₁₈ X_16,10,17,9 X_20,13,21,14 X_12,19,13,20 X_22,16,7,15 X_4,22,5,21
Gauss code	{1, -4, 3, -11, 5, -6}, {6, -1, 7, -3, 2, -9, 8, -5, 10, -7, 4, -2, 9, -8, 11, -10}

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {u^{2}v^{6}-3u^{2}v^{5}+4u^{2}v^{4}-4u^{2}v^{3}+3u^{2}v^{2}-u^{2}v-uv^{6}+4uv^{5}-7uv^{4}+7uv^{3}-7uv^{2}+4uv-u-v^{5}+3v^{4}-4v^{3}+4v^{2}-3v+1}{uv^{3}}}$ (db)
Jones polynomial	$-q^{5/2}+4q^{3/2}-8{\sqrt {q}}+{\frac {12}{\sqrt {q}}}-{\frac {18}{q^{3/2}}}+{\frac {19}{q^{5/2}}}-{\frac {20}{q^{7/2}}}+{\frac {18}{q^{9/2}}}-{\frac {13}{q^{11/2}}}+{\frac {8}{q^{13/2}}}-{\frac {4}{q^{15/2}}}+{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-a^{5}z^{7}-4a^{5}z^{5}-4a^{5}z^{3}+a^{3}z^{9}+6a^{3}z^{7}+12a^{3}z^{5}+8a^{3}z^{3}+a^{3}z^{-1}-az^{7}-4az^{5}-4az^{3}-az-az^{-1}$ (db)
Kauffman polynomial	$a^{10}z^{4}+4a^{9}z^{5}-2a^{9}z^{3}+8a^{8}z^{6}-7a^{8}z^{4}+2a^{8}z^{2}+11a^{7}z^{7}-13a^{7}z^{5}+4a^{7}z^{3}+11a^{6}z^{8}-14a^{6}z^{6}+4a^{6}z^{2}+8a^{5}z^{9}-9a^{5}z^{7}-4a^{5}z^{5}+2a^{5}z^{3}+3a^{4}z^{10}+7a^{4}z^{8}-30a^{4}z^{6}+18a^{4}z^{4}+14a^{3}z^{9}-42a^{3}z^{7}+38a^{3}z^{5}-14a^{3}z^{3}+a^{3}z-a^{3}z^{-1}+3a^{2}z^{10}-22a^{2}z^{6}+23a^{2}z^{4}-4a^{2}z^{2}+a^{2}+6az^{9}-21az^{7}+z^{7}a^{-1}+22az^{5}-3z^{5}a^{-1}-8az^{3}+2z^{3}a^{-1}+az-az^{-1}+4z^{8}-14z^{6}+13z^{4}-2z^{2}$ (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		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

3

-3

2

5

1

4

0

7

3

-4

-2

11

5

6

-4

9

8

-1

-6

11

10

1

-8

8

10

2

-10

5

10

-5

-12

3

8

5

-14

1

5

-4

-16

3

-18

1

-1

Integral Khovanov Homology

(db, data source)

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