L7a1

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	[edit L7a1 Quick Notes] L7a1 is $7_{6}^{2}$ in the Rolfsen table of links.

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Link Presentations

[edit Notes on L7a1's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_10,6,11,5 X₈₄₉₃ X_14,10,5,9 X_2,12,3,11
Gauss code	{1, -7, 5, -3}, {4, -1, 2, -5, 6, -4, 7, -2, 3, -6}

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

Polynomial invariants

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

-3

-2

-1

0

1

2

3

4

χ

10

1

8

2

-2

6

2

1

4

2

0

2

3

2

1

0

2

4

2

-2

1

0

-4

2

-6

1

-1

Integral Khovanov Homology

(db, data source)

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