L7a7

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	[edit L7a7 Quick Notes] L7a7 is $7_{1}^{3}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L7a7's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvw+u(-v)-uw+2u-2vw+v+w-1}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}}}$ (db)
Jones polynomial	$q^{-4}-q^{3}-q^{-3}+3q^{2}+4q^{-2}-3q-3q^{-1}+4$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{4}z^{-2}+a^{4}-2z^{2}a^{2}-2a^{2}z^{-2}-3a^{2}+z^{4}+2z^{2}+z^{-2}+2-z^{2}a^{-2}$ (db)
Kauffman polynomial	$a^{2}z^{6}+z^{6}+a^{3}z^{5}+4az^{5}+3z^{5}a^{-1}+a^{4}z^{4}+a^{2}z^{4}+3z^{4}a^{-2}+3z^{4}-4az^{3}-3z^{3}a^{-1}+z^{3}a^{-3}-3a^{4}z^{2}-5a^{2}z^{2}-3z^{2}a^{-2}-5z^{2}-3a^{3}z-3az+3a^{4}+5a^{2}+3+2a^{3}z^{-1}+2az^{-1}-a^{4}z^{-2}-2a^{2}z^{-2}-z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

χ

7

1

-1

5

2

3

1

0

1

3

2

1

-1

3

4

1

-3

1

-5

3

-7

1

0

-9

1

Integral Khovanov Homology

(db, data source)

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