L10n109

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_19,11,20,14 X_13,15,14,20 X_9,18,10,19 X_11,10,12,5 X_17,1,18,4
Gauss code	{1, -4, -3, 10}, {-9, 2, -7, 6}, {-2, -1, 5, 3, -8, 9}, {4, -5, -10, 8, -6, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(2)t(3)^{2}+t(2)t(4)t(3)^{2}-t(4)t(3)^{2}+t(1)t(4)^{2}t(3)-t(1)t(2)t(4)^{2}t(3)+t(2)t(3)-t(1)t(4)t(3)+2t(1)t(2)t(4)t(3)-t(2)t(4)t(3)+2t(4)t(3)-t(3)-t(1)t(4)^{2}+t(1)t(4)-t(1)t(2)t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$-2q^{3/2}+2{\sqrt {q}}-{\frac {6}{\sqrt {q}}}+{\frac {4}{q^{3/2}}}-{\frac {7}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {4}{q^{9/2}}}+{\frac {2}{q^{11/2}}}-{\frac {1}{q^{13/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{3}+a^{5}z^{-3}+2a^{5}z+2a^{5}z^{-1}-a^{3}z^{5}-4a^{3}z^{3}-3a^{3}z^{-3}-8a^{3}z-7a^{3}z^{-1}+3az^{3}+3az^{-3}-a^{-1}z^{-3}+8az+8az^{-1}-2za^{-1}-3a^{-1}z^{-1}$ (db)
Kauffman polynomial	$a^{7}z^{5}-3a^{7}z^{3}+a^{7}z+2a^{6}z^{6}-5a^{6}z^{4}+a^{6}z^{2}+3a^{5}z^{7}-11a^{5}z^{5}+15a^{5}z^{3}-a^{5}z^{-3}-12a^{5}z+5a^{5}z^{-1}+a^{4}z^{8}+a^{4}z^{6}-11a^{4}z^{4}+17a^{4}z^{2}+3a^{4}z^{-2}-10a^{4}+5a^{3}z^{7}-20a^{3}z^{5}+36a^{3}z^{3}-3a^{3}z^{-3}-29a^{3}z+12a^{3}z^{-1}+a^{2}z^{8}-8a^{2}z^{4}+23a^{2}z^{2}+6a^{2}z^{-2}-19a^{2}+2az^{7}-8az^{5}+21az^{3}+3z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-23az-7za^{-1}+12az^{-1}+5a^{-1}z^{-1}+z^{6}-2z^{4}+7z^{2}+3z^{-2}-10$ (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

4

2

1

0

5

1

4

-2

3

5

2

-4

4

1

3

-6

2

5

3

-8

2

0

-10

2

-12

1

2

-1

-14

1

Integral Khovanov Homology

(db, data source)

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