L10n28

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_7,14,8,15 X_11,19,12,18 X_15,20,16,5 X_19,16,20,17 X_17,13,18,12 X_13,8,14,9 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, -3, 8, 10, -2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{3}+t(2)^{3}-2t(1)t(2)^{2}-t(2)^{2}-t(1)t(2)-2t(2)+t(1)+1}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {1}{\sqrt {q}}}-{\frac {2}{q^{3/2}}}+{\frac {1}{q^{5/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}-{\frac {1}{q^{13/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{17/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-1}-2a^{7}z-2a^{7}z^{-1}+a^{5}z^{-1}+a^{3}z^{5}+4a^{3}z^{3}+3a^{3}z+a^{3}z^{-1}-az^{3}-3az-az^{-1}$ (db)
Kauffman polynomial	$-z^{7}a^{9}+6z^{5}a^{9}-10z^{3}a^{9}+6za^{9}-a^{9}z^{-1}-z^{8}a^{8}+6z^{6}a^{8}-9z^{4}a^{8}+4z^{2}a^{8}-a^{8}-2z^{7}a^{7}+14z^{5}a^{7}-26z^{3}a^{7}+14za^{7}-2a^{7}z^{-1}-z^{8}a^{6}+7z^{6}a^{6}-13z^{4}a^{6}+8z^{2}a^{6}-3a^{6}-z^{7}a^{5}+7z^{5}a^{5}-13z^{3}a^{5}+8za^{5}-a^{5}z^{-1}-z^{4}a^{4}+4z^{2}a^{4}-2a^{4}-2z^{5}a^{3}+7z^{3}a^{3}-4za^{3}+a^{3}z^{-1}-z^{6}a^{2}+3z^{4}a^{2}-a^{2}-z^{5}a+4z^{3}a-4za+az^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

χ

2

1

0

-2

2

1

-4

1

-6

1

0

-8

1

2

1

0

-10

1

-1

-12

1

2

1

-14

1

0

-16

0

-18

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-4$	$i=-2$	$i=0$
$r=-8$		${\mathbb {Z} }$
$r=-7$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-6$		${\mathbb {Z} }$
$r=-5$		${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-1$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{2}$
$r=1$	${\mathbb {Z} }$
$r=2$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$