L10a66

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X₆₇₁₈ X_16,11,17,12 X_14,6,15,5 X_4,16,5,15 X_20,17,7,18 X_18,14,19,13 X_12,20,13,19
Gauss code	{1, -2, 3, -7, 6, -4}, {4, -1, 2, -3, 5, -10, 9, -6, 7, -5, 8, -9, 10, -8}

A Braid Representative

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Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)^{2}t(2)^{4}+t(1)t(2)^{4}+3t(1)^{2}t(2)^{3}-3t(1)t(2)^{3}+t(2)^{3}-3t(1)^{2}t(2)^{2}+3t(1)t(2)^{2}-3t(2)^{2}+t(1)^{2}t(2)-3t(1)t(2)+3t(2)+t(1)-1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$-q^{5/2}+2q^{3/2}-4{\sqrt {q}}+{\frac {6}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {8}{q^{5/2}}}-{\frac {9}{q^{7/2}}}+{\frac {7}{q^{9/2}}}-{\frac {5}{q^{11/2}}}+{\frac {3}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{5}z^{5}+3a^{5}z^{3}+2a^{5}z+a^{5}z^{-1}-a^{3}z^{7}-5a^{3}z^{5}-9a^{3}z^{3}-8a^{3}z-2a^{3}z^{-1}+2az^{5}+8az^{3}-z^{3}a^{-1}+8az-3za^{-1}+2az^{-1}-a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-z^{3}a^{9}-3z^{4}a^{8}+z^{2}a^{8}-5z^{5}a^{7}+4z^{3}a^{7}-za^{7}-6z^{6}a^{6}+7z^{4}a^{6}-z^{2}a^{6}-6z^{7}a^{5}+11z^{5}a^{5}-5z^{3}a^{5}+2za^{5}-a^{5}z^{-1}-4z^{8}a^{4}+7z^{6}a^{4}-z^{9}a^{3}-6z^{7}a^{3}+29z^{5}a^{3}-31z^{3}a^{3}+12za^{3}-2a^{3}z^{-1}-6z^{8}a^{2}+22z^{6}a^{2}-22z^{4}a^{2}+7z^{2}a^{2}-a^{2}-z^{9}a-z^{7}a+18z^{5}a-29z^{3}a+14za-2az^{-1}-2z^{8}+9z^{6}-12z^{4}+5z^{2}-z^{7}a^{-1}+5z^{5}a^{-1}-8z^{3}a^{-1}+5za^{-1}-a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

6

1

4

1

-1

2

3

1

2

0

3

1

-2

5

3

2

-4

4

0

-6

5

4

1

-8

3

5

2

-10

2

4

-2

-12

1

3

2

-14

2

-2

-16

1

Integral Khovanov Homology

(db, data source)

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