L10a78

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}-4t(1)^{2}t(2)^{3}+4t(1)t(2)^{3}+4t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+4t(2)^{2}+4t(1)t(2)-4t(2)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	${\frac {8}{q^{9/2}}}-{\frac {11}{q^{7/2}}}-q^{5/2}+{\frac {10}{q^{5/2}}}+3q^{3/2}-{\frac {10}{q^{3/2}}}-{\frac {1}{q^{15/2}}}+{\frac {3}{q^{13/2}}}-{\frac {6}{q^{11/2}}}-5{\sqrt {q}}+{\frac {8}{\sqrt {q}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+3a^{5}z^{3}+3a^{5}z+2a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-10a^{3}z^{3}-10a^{3}z-3a^{3}z^{-1}+2az^{5}+7az^{3}-z^{3}a^{-1}+6az-2za^{-1}+az^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{9}-3z^{4}a^{8}-6z^{5}a^{7}+5z^{3}a^{7}-2za^{7}-8z^{6}a^{6}+10z^{4}a^{6}-2z^{2}a^{6}-9z^{7}a^{5}+19z^{5}a^{5}-13z^{3}a^{5}+7za^{5}-2a^{5}z^{-1}-6z^{8}a^{4}+10z^{6}a^{4}+4z^{4}a^{4}-7z^{2}a^{4}+3a^{4}-2z^{9}a^{3}-5z^{7}a^{3}+31z^{5}a^{3}-31z^{3}a^{3}+12za^{3}-3a^{3}z^{-1}-9z^{8}a^{2}+31z^{6}a^{2}-26z^{4}a^{2}+z^{2}a^{2}+3a^{2}-2z^{9}a+3z^{7}a+10z^{5}a-17z^{3}a+5za-az^{-1}-3z^{8}+13z^{6}-17z^{4}+6z^{2}+1-z^{7}a^{-1}+4z^{5}a^{-1}-5z^{3}a^{-1}+2za^{-1}$ (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

6

1

4

2

-2

2

3

1

2

0

5

2

-3

-2

5

3

2

-4

6

0

-6

5

4

1

-8

3

6

3

-10

3

5

-2

-12

3

-14

1

3

-2

-16

1

Integral Khovanov Homology

(db, data source)

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