L10n10

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_14,7,15,8 X_4,15,1,16 X_5,12,6,13 X₃₈₄₉ X_9,16,10,17 X_17,20,18,5 X_11,19,12,18 X_19,11,20,10 X_13,2,14,3
Gauss code	{1, 10, -5, -3}, {-4, -1, 2, 5, -6, 9, -8, 4, -10, -2, 3, 6, -7, 8, -9, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(2)^{5}-2t(1)t(2)^{4}-2t(2)+1}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-{\frac {2}{q^{5/2}}}+{\frac {1}{q^{3/2}}}-{\frac {1}{q^{17/2}}}+{\frac {1}{q^{15/2}}}-{\frac {1}{q^{13/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{\sqrt {q}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{9}z+a^{9}z^{-1}-a^{7}z^{5}-5a^{7}z^{3}-6a^{7}z-3a^{7}z^{-1}+a^{5}z^{7}+6a^{5}z^{5}+11a^{5}z^{3}+9a^{5}z+4a^{5}z^{-1}-a^{3}z^{5}-5a^{3}z^{3}-6a^{3}z-2a^{3}z^{-1}$ (db)
Kauffman polynomial	$-za^{11}-z^{2}a^{10}+a^{10}-z^{3}a^{9}+2za^{9}-a^{9}z^{-1}-z^{6}a^{8}+4z^{4}a^{8}-4z^{2}a^{8}+3a^{8}-2z^{7}a^{7}+11z^{5}a^{7}-18z^{3}a^{7}+12za^{7}-3a^{7}z^{-1}-z^{8}a^{6}+4z^{6}a^{6}-z^{4}a^{6}-5z^{2}a^{6}+3a^{6}-3z^{7}a^{5}+17z^{5}a^{5}-28z^{3}a^{5}+17za^{5}-4a^{5}z^{-1}-z^{8}a^{4}+5z^{6}a^{4}-5z^{4}a^{4}-2z^{2}a^{4}+2a^{4}-z^{7}a^{3}+6z^{5}a^{3}-11z^{3}a^{3}+8za^{3}-2a^{3}z^{-1}$ (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

χ

0

1

-2

0

-4

2

1

-6

1

-8

2

1

-10

1

2

1

0

-12

1

2

-1

-14

1

0

-16

1

0

-18

1

Integral Khovanov Homology

(db, data source)

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