L10n105

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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_7,17,8,16 X_14,9,11,10 X_20,13,15,14 X_19,5,20,10 X_11,18,12,19 X_17,1,18,4
Gauss code	{1, -4, -3, 10}, {-9, 2, 7, -6}, {-2, -1, -5, 3, 6, 8}, {4, 5, -10, 9, -8, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

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

-4

-3

-2

-1

0

1

2

χ

4

2

1

2

1

0

3

-2

1

3

2

-4

3

1

2

-6

3

-8

2

1

-10

2

Integral Khovanov Homology

(db, data source)

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