L11n166

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Link Presentations

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Planar diagram presentation	X₈₁₉₂ X_2,9,3,10 X_10,3,11,4 X_14,5,15,6 X_11,19,12,18 X_19,7,20,22 X_15,21,16,20 X_21,17,22,16 X_17,13,18,12 X₆₇₁₈ X_4,13,5,14
Gauss code	{1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, -5, 9, 11, -4, -7, 8, -9, 5, -6, 7, -8, 6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {t(1)^{2}t(2)^{4}+t(1)t(2)^{4}-t(1)^{2}t(2)^{3}-t(1)t(2)^{3}+t(2)^{3}+t(1)t(2)^{2}+t(1)^{2}t(2)-t(1)t(2)-t(2)+t(1)+1}{t(1)t(2)^{2}}}$ (db)
Jones polynomial	$q^{7/2}-q^{5/2}+q^{3/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}+{\frac {1}{q^{3/2}}}-{\frac {2}{q^{5/2}}}+{\frac {2}{q^{7/2}}}-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$-az^{7}-7az^{5}+z^{5}a^{-1}+a^{5}z^{3}+a^{3}z^{3}-15az^{3}+5z^{3}a^{-1}+2a^{5}z+a^{3}z-12az+6za^{-1}+a^{5}z^{-1}-2az^{-1}+a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-az^{9}-z^{9}a^{-1}-a^{2}z^{8}-z^{8}a^{-2}-2z^{8}-a^{5}z^{7}+8az^{7}+7z^{7}a^{-1}-a^{6}z^{6}-a^{4}z^{6}+8a^{2}z^{6}+7z^{6}a^{-2}+15z^{6}-a^{7}z^{5}+4a^{5}z^{5}-20az^{5}-15z^{5}a^{-1}+3a^{6}z^{4}+3a^{4}z^{4}-18a^{2}z^{4}-15z^{4}a^{-2}-33z^{4}+4a^{7}z^{3}-5a^{5}z^{3}+22az^{3}+13z^{3}a^{-1}-a^{6}z^{2}-2a^{4}z^{2}+13a^{2}z^{2}+11z^{2}a^{-2}+25z^{2}-3a^{7}z+4a^{5}z-13az-6za^{-1}+a^{4}-3a^{2}-2a^{-2}-5-a^{5}z^{-1}+2az^{-1}+a^{-1}z^{-1}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

3

4

5

χ

8

1

-1

6

0

4

1

0

2

1

0

1

2

-2

2

3

1

0

-4

1

-6

1

2

1

0

-8

1

0

-10

1

-12

1

0

-14

1

Integral Khovanov Homology

(db, data source)

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