L10a163

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_20,10,13,9 X_6,20,1,19 X_18,7,19,8 X_4,11,5,12 X_16,6,17,5 X_10,16,11,15 X_12,17,7,18 X_2,13,3,14 X_14,3,15,4
Gauss code	{1, -9, 10, -5, 6, -3}, {4, -1, 2, -7, 5, -8}, {9, -10, 7, -6, 8, -4, 3, -2}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(v-1)(w-1)\left(uvw^{2}-uvw+uw+vw-w+1\right)}{uvw^{3/2}}}$ (db)
Jones polynomial	$q^{-7}-4q^{-6}+8q^{-5}-12q^{-4}+q^{3}+16q^{-3}-4q^{2}-15q^{-2}+8q+16q^{-1}-11$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$-a^{2}z^{8}+a^{4}z^{6}-5a^{2}z^{6}+z^{6}+3a^{4}z^{4}-8a^{2}z^{4}+3z^{4}+2a^{4}z^{2}-4a^{2}z^{2}+2z^{2}-a^{2}+1+a^{4}z^{-2}-2a^{2}z^{-2}+z^{-2}$ (db)
Kauffman polynomial	$a^{8}z^{4}+4a^{7}z^{5}-2a^{7}z^{3}+8a^{6}z^{6}-8a^{6}z^{4}+3a^{6}z^{2}+10a^{5}z^{7}-12a^{5}z^{5}+4a^{5}z^{3}+8a^{4}z^{8}-7a^{4}z^{6}-3a^{4}z^{4}+2a^{4}z^{2}-a^{4}z^{-2}+a^{4}+3a^{3}z^{9}+9a^{3}z^{7}-26a^{3}z^{5}+12a^{3}z^{3}-a^{3}z+2a^{3}z^{-1}+14a^{2}z^{8}-31a^{2}z^{6}+z^{6}a^{-2}+18a^{2}z^{4}-2z^{4}a^{-2}-4a^{2}z^{2}+z^{2}a^{-2}-2a^{2}z^{-2}+a^{2}+3az^{9}+3az^{7}+4z^{7}a^{-1}-20az^{5}-10z^{5}a^{-1}+12az^{3}+6z^{3}a^{-1}-az+2az^{-1}+6z^{8}-15z^{6}+10z^{4}-2z^{2}-z^{-2}+1$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

3

-3

3

5

1

4

1

6

3

-3

-1

10

5

-3

7

8

1

-5

9

8

1

-7

5

9

4

-9

3

7

-4

-11

1

5

4

-13

3

-3

-15

1

Integral Khovanov Homology

(db, data source)

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