L7a5

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	[edit L7a5 Quick Notes] L7a5 is $7_{2}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L7a5's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_10,3,11,4 X_12,6,13,5 X_14,11,7,12 X_4,14,5,13 X₂₇₃₈ X_6,9,1,10
Gauss code	{1, -6, 2, -5, 3, -7}, {6, -1, 7, -2, 4, -3, 5, -4}

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

Polynomial invariants

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

-5

-4

-3

-2

-1

0

1

2

χ

4

1

2

1

-1

0

2

1

-2

2

0

-4

2

1

-6

1

3

2

-8

1

0

-10

1

-12

1

-1

Integral Khovanov Homology

(db, data source)

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