L10n103

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_14,7,15,8 X_8,13,5,14 X_11,18,12,19 X_20,16,17,15 X_16,20,9,19 X_17,12,18,13 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 3, -4}, {-8, 5, 7, -6}, {10, -2, -5, 8, 4, -3, 6, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)(t(3)-1)(t(4)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}}}$ (db)
Jones polynomial	$-{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {5}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {4}{q^{9/2}}}-{\frac {6}{q^{11/2}}}+{\frac {1}{q^{13/2}}}-{\frac {1}{q^{15/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z^{-3}-3a^{7}z^{-3}-3a^{7}z^{-1}-a^{5}z^{3}+3a^{5}z^{-3}+2a^{5}z+6a^{5}z^{-1}+a^{3}z^{5}+2a^{3}z^{3}-a^{3}z^{-3}-a^{3}z-3a^{3}z^{-1}-az^{3}-az$ (db)
Kauffman polynomial	$a^{9}z^{3}-a^{9}z^{-3}-3a^{9}z+3a^{9}z^{-1}+a^{8}z^{4}+2a^{8}z^{2}+3a^{8}z^{-2}-6a^{8}+a^{7}z^{7}-4a^{7}z^{5}+12a^{7}z^{3}-3a^{7}z^{-3}-9a^{7}z+6a^{7}z^{-1}+a^{6}z^{8}-2a^{6}z^{6}+a^{6}z^{4}+9a^{6}z^{2}+6a^{6}z^{-2}-11a^{6}+4a^{5}z^{7}-12a^{5}z^{5}+16a^{5}z^{3}-3a^{5}z^{-3}-11a^{5}z+6a^{5}z^{-1}+a^{4}z^{8}+a^{4}z^{6}-7a^{4}z^{4}+8a^{4}z^{2}+3a^{4}z^{-2}-6a^{4}+3a^{3}z^{7}-7a^{3}z^{5}+3a^{3}z^{3}-a^{3}z^{-3}-4a^{3}z+3a^{3}z^{-1}+3a^{2}z^{6}-7a^{2}z^{4}+a^{2}z^{2}+az^{5}-2az^{3}+az$ (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

2

1

0

2

-2

3

1

2

-4

3

4

1

-6

4

1

3

-8

1

3

-10

6

4

2

-12

1

6

5

-14

0

-16

1

Integral Khovanov Homology

(db, data source)

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