L10a156

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(3)-1)(-t(2)t(1)+t(2)t(3)t(1)+t(1)+t(2)-1)(-t(1)t(3)+t(1)t(2)t(3)-t(2)t(3)+t(3)-1)}{t(1)t(2)t(3)^{3/2}}}$ (db)
Jones polynomial	$-q^{5}-q^{-5}+4q^{4}+4q^{-4}-8q^{3}-8q^{-3}+13q^{2}+13q^{-2}-15q-15q^{-1}+18$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$z^{8}-a^{2}z^{6}-z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-3z^{4}a^{-2}+8z^{4}-2a^{2}z^{2}-2z^{2}a^{-2}+3z^{2}+a^{2}+a^{-2}-2+a^{2}z^{-2}+a^{-2}z^{-2}-2z^{-2}$ (db)
Kauffman polynomial	$a^{5}z^{5}+z^{5}a^{-5}-a^{5}z^{3}-z^{3}a^{-5}+4a^{4}z^{6}+4z^{6}a^{-4}-6a^{4}z^{4}-6z^{4}a^{-4}+2a^{4}z^{2}+2z^{2}a^{-4}+7a^{3}z^{7}+7z^{7}a^{-3}-11a^{3}z^{5}-11z^{5}a^{-3}+4a^{3}z^{3}+4z^{3}a^{-3}+7a^{2}z^{8}+7z^{8}a^{-2}-9a^{2}z^{6}-9z^{6}a^{-2}+2a^{2}z^{4}+2z^{4}a^{-2}+a^{2}z^{2}+z^{2}a^{-2}+a^{2}z^{-2}+a^{-2}z^{-2}-2a^{2}-2a^{-2}+3az^{9}+3z^{9}a^{-1}+7az^{7}+7z^{7}a^{-1}-19az^{5}-19z^{5}a^{-1}+9az^{3}+9z^{3}a^{-1}+2az+2za^{-1}-2az^{-1}-2a^{-1}z^{-1}+14z^{8}-26z^{6}+16z^{4}-2z^{2}+2z^{-2}-3$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

11

1

-1

9

3

7

5

1

-4

5

8

3

5

3

8

6

-2

1

10

7

3

-1

7

10

3

-3

6

8

-2

-5

3

8

5

-7

1

5

-4

-9

3

-11

1

-1

Integral Khovanov Homology

(db, data source)

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