L8a7

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	[edit L8a7 Quick Notes] L8a7 is $8_{14}^{2}$ in the Rolfsen table of links.

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_10,5,11,6 X_12,3,13,4 X_16,11,5,12 X_2,9,3,10 X_8,13,9,14
Gauss code	{1, -7, 5, -3}, {4, -1, 2, -8, 7, -4, 6, -5, 8, -2, 3, -6}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(2)^{3}+4t(1)t(2)^{2}-4t(2)^{2}-4t(1)t(2)+4t(2)+t(1)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {5}{q^{9/2}}}-{\frac {6}{q^{7/2}}}+{\frac {3}{q^{5/2}}}-{\frac {1}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {3}{q^{17/2}}}-{\frac {4}{q^{15/2}}}+{\frac {6}{q^{13/2}}}-{\frac {7}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z-a^{9}z^{-1}-a^{7}z^{3}+2a^{7}z+3a^{7}z^{-1}-3a^{5}z^{3}-5a^{5}z-2a^{5}z^{-1}-a^{3}z^{3}$ (db)
Kauffman polynomial	$a^{11}z^{5}-2a^{11}z^{3}+a^{11}z+3a^{10}z^{6}-8a^{10}z^{4}+5a^{10}z^{2}+a^{10}+2a^{9}z^{7}-a^{9}z^{5}-5a^{9}z^{3}+2a^{9}z-a^{9}z^{-1}+8a^{8}z^{6}-16a^{8}z^{4}+4a^{8}z^{2}+3a^{8}+2a^{7}z^{7}+4a^{7}z^{5}-12a^{7}z^{3}+6a^{7}z-3a^{7}z^{-1}+5a^{6}z^{6}-5a^{6}z^{4}-a^{6}z^{2}+3a^{6}+6a^{5}z^{5}-8a^{5}z^{3}+5a^{5}z-2a^{5}z^{-1}+3a^{4}z^{4}+a^{3}z^{3}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

1

-4

3

1

-2

-6

3

-8

2

3

1

-10

5

3

2

-12

2

3

1

-14

2

4

-2

-16

1

2

1

-18

2

-2

-20

1

Integral Khovanov Homology

(db, data source)

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