L10a84

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_14,9,15,10 X₄₇₅₈ X_16,6,17,5 X_18,16,19,15 X_6,18,1,17 X_20,11,7,12 X_10,19,11,20 X_2,14,3,13 X_12,4,13,3
Gauss code	{1, -9, 10, -3, 4, -6}, {3, -1, 2, -8, 7, -10, 9, -2, 5, -4, 6, -5, 8, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-2t(1)t(2)^{4}+t(2)^{4}-2t(1)^{2}t(2)^{3}+6t(1)t(2)^{3}-3t(2)^{3}+3t(1)^{2}t(2)^{2}-7t(1)t(2)^{2}+3t(2)^{2}-3t(1)^{2}t(2)+6t(1)t(2)-2t(2)+t(1)^{2}-2t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{9/2}-{\frac {3}{q^{9/2}}}-4q^{7/2}+{\frac {7}{q^{7/2}}}+7q^{5/2}-{\frac {11}{q^{5/2}}}-11q^{3/2}+{\frac {13}{q^{3/2}}}+{\frac {1}{q^{11/2}}}+13{\sqrt {q}}-{\frac {15}{\sqrt {q}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-a^{3}z^{5}-3a^{3}z^{3}+z^{3}a^{-3}-4a^{3}z+za^{-3}-a^{3}z^{-1}+az^{7}+5az^{5}-2z^{5}a^{-1}+11az^{3}-6z^{3}a^{-1}+10az-6za^{-1}+3az^{-1}-2a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{6}z^{4}-a^{6}z^{2}+3a^{5}z^{5}-2a^{5}z^{3}+6a^{4}z^{6}+z^{6}a^{-4}-7a^{4}z^{4}-2z^{4}a^{-4}+5a^{4}z^{2}+z^{2}a^{-4}-a^{4}+8a^{3}z^{7}+4z^{7}a^{-3}-13a^{3}z^{5}-11z^{5}a^{-3}+13a^{3}z^{3}+8z^{3}a^{-3}-6a^{3}z-za^{-3}+a^{3}z^{-1}+6a^{2}z^{8}+5z^{8}a^{-2}-4a^{2}z^{6}-11z^{6}a^{-2}-7a^{2}z^{4}+3z^{4}a^{-2}+10a^{2}z^{2}+2z^{2}a^{-2}-3a^{2}+2az^{9}+2z^{9}a^{-1}+11az^{7}+7z^{7}a^{-1}-35az^{5}-30z^{5}a^{-1}+33az^{3}+26z^{3}a^{-1}-15az-10za^{-1}+3az^{-1}+2a^{-1}z^{-1}+11z^{8}-22z^{6}+6z^{4}+5z^{2}-3$ (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

10

1

-1

8

3

6

4

1

-3

4

7

3

4

2

7

5

-2

0

8

6

2

-2

6

8

2

-4

5

7

-2

-6

2

6

4

-8

1

5

-4

-10

2

-12

1

-1

Integral Khovanov Homology

(db, data source)

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