L7a4

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	[edit L7a4 Quick Notes] L7a4 is $7_{3}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L7a4's Link Presentations]

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

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

Polynomial invariants

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

-2

-1

0

1

2

3

4

5

χ

12

1

-1

10

1

8

1

0

6

2

1

4

1

0

2

0

1

3

2

-2

0

-4

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=0$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=2}
$r=-2$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }^{2}$
$r=1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=4$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=5$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$