L7a2

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	[edit L7a2 Quick Notes] L7a2 is $7_{5}^{2}$ in the Rolfsen table of links.

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Link Presentations

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

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

Polynomial invariants

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

χ

-2

1

-4

2

1

-1

-6

2

-8

1

2

1

-10

3

2

1

-12

1

2

1

-14

1

2

-1

-16

1

-18

1

-1

Integral Khovanov Homology

(db, data source)

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