L7a6

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	[edit L7a6 Quick Notes] L7a6 is $7_{1}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L7a6's Link Presentations]

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

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

14

1

-1

12

1

10

1

0

8

2

1

6

1

2

1

4

1

0

2

1

2

1

0

-2

1

Integral Khovanov Homology

(db, data source)

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