L7a3

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	[edit L7a3 Quick Notes] L7a3 is $7_{4}^{2}$ in the Rolfsen table of links.

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Link Presentations

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

14

1

-1

12

1

10

2

1

-1

8

1

0

6

1

2

1

4

2

1

2

1

3

2

0

-2

1

Integral Khovanov Homology

(db, data source)

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