L10a67

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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_14,5,15,6 X_18,11,19,12 X_20,16,7,15 X_16,20,17,19 X_12,17,13,18 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -10, 4, -9}, {9, -1, 2, -3, 5, -8, 10, -4, 6, -7, 8, -5, 7, -6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)^{2}t(2)^{4}-4t(1)^{2}t(2)^{3}+2t(1)t(2)^{3}+2t(1)^{2}t(2)^{2}-5t(1)t(2)^{2}+2t(2)^{2}+2t(1)t(2)-4t(2)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-{\frac {7}{q^{9/2}}}+{\frac {5}{q^{7/2}}}-{\frac {4}{q^{5/2}}}+{\frac {2}{q^{3/2}}}-{\frac {1}{q^{21/2}}}+{\frac {2}{q^{19/2}}}-{\frac {4}{q^{17/2}}}+{\frac {6}{q^{15/2}}}-{\frac {7}{q^{13/2}}}+{\frac {7}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$z^{3}a^{9}+3za^{9}+a^{9}z^{-1}-2z^{5}a^{7}-8z^{3}a^{7}-8za^{7}-2a^{7}z^{-1}+z^{7}a^{5}+5z^{5}a^{5}+8z^{3}a^{5}+6za^{5}+2a^{5}z^{-1}-z^{5}a^{3}-4z^{3}a^{3}-4za^{3}-a^{3}z^{-1}$ (db)
Kauffman polynomial	$a^{13}z^{3}-a^{13}z+2a^{12}z^{4}-a^{12}z^{2}+3a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+4a^{10}z^{6}-6a^{10}z^{4}+4a^{10}z^{2}+4a^{9}z^{7}-8a^{9}z^{5}+6a^{9}z^{3}-4a^{9}z+a^{9}z^{-1}+3a^{8}z^{8}-6a^{8}z^{6}+a^{8}z^{4}+a^{7}z^{9}+3a^{7}z^{7}-20a^{7}z^{5}+24a^{7}z^{3}-12a^{7}z+2a^{7}z^{-1}+5a^{6}z^{8}-19a^{6}z^{6}+20a^{6}z^{4}-8a^{6}z^{2}+a^{6}+a^{5}z^{9}-14a^{5}z^{5}+23a^{5}z^{3}-11a^{5}z+2a^{5}z^{-1}+2a^{4}z^{8}-9a^{4}z^{6}+11a^{4}z^{4}-3a^{4}z^{2}+a^{3}z^{7}-5a^{3}z^{5}+8a^{3}z^{3}-5a^{3}z+a^{3}z^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

0

1

-2

1

-1

-4

3

1

2

-6

3

2

-1

-8

4

2

-10

3

0

-12

4

0

-14

2

3

1

-16

2

4

-2

-18

1

3

2

-20

1

-1

-22

1

Integral Khovanov Homology

(db, data source)

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