L8a18

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	[edit L8a18 Quick Notes] L8a18 is $8_{1}^{3}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L8a18's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {uv^{2}w^{2}-uvw^{2}+uvw-uw+u-v^{2}w^{2}+v^{2}w-vw+v-1}{{\sqrt {u}}vw}}$ (db)
Jones polynomial	$q^{8}-2q^{7}+3q^{6}-3q^{5}+4q^{4}-2q^{3}+3q^{2}-q+1$ (db)
Signature	4 (db)
HOMFLY-PT polynomial	$z^{4}a^{-6}+3z^{2}a^{-6}+a^{-6}z^{-2}+2a^{-6}-z^{6}a^{-4}-5z^{4}a^{-4}-8z^{2}a^{-4}-2a^{-4}z^{-2}-6a^{-4}+z^{4}a^{-2}+4z^{2}a^{-2}+a^{-2}z^{-2}+4a^{-2}$ (db)
Kauffman polynomial	$z^{2}a^{-10}+2z^{3}a^{-9}+3z^{4}a^{-8}-3z^{2}a^{-8}+a^{-8}+3z^{5}a^{-7}-4z^{3}a^{-7}+3z^{6}a^{-6}-8z^{4}a^{-6}+6z^{2}a^{-6}+a^{-6}z^{-2}-3a^{-6}+z^{7}a^{-5}-7z^{3}a^{-5}+6za^{-5}-2a^{-5}z^{-1}+4z^{6}a^{-4}-16z^{4}a^{-4}+18z^{2}a^{-4}+2a^{-4}z^{-2}-8a^{-4}+z^{7}a^{-3}-3z^{5}a^{-3}-z^{3}a^{-3}+6za^{-3}-2a^{-3}z^{-1}+z^{6}a^{-2}-5z^{4}a^{-2}+8z^{2}a^{-2}+a^{-2}z^{-2}-5a^{-2}$ (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		\

-2

-1

0

1

2

3

4

5

6

χ

17

1

15

2

1

-1

13

1

11

2

0

9

2

1

7

1

3

2

5

2

1

3

1

3

2

1

0

-1

1

Integral Khovanov Homology

(db, data source)

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