L10a36

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,12,15,11 X_20,15,5,16 X_16,7,17,8 X_8,19,9,20 X_18,9,19,10 X_10,17,11,18 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, 5, -6, 7, -8, 3, -2, 10, -3, 4, -5, 8, -7, 6, -4}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

χ

6

1

-1

4

3

2

4

1

-3

0

6

3

-2

7

6

-1

-4

6

4

2

-6

4

7

3

-8

4

6

-2

-10

1

4

3

-12

1

4

-3

-14

1

-16

1

-1

Integral Khovanov Homology

(db, data source)

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