L10n106

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_5,12,6,13 X₃₈₄₉ X_15,2,16,3 X_16,7,17,8 X_9,11,10,14 X_13,15,14,20 X_19,5,20,10 X_11,18,12,19 X_4,17,1,18
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 $u$ , $v$ , $w$ , ...)	$-{\frac {(t(2)-1)(t(3)-1)(t(4)-1)(t(1)t(3)t(4)-1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {3}{\sqrt {q}}}-{\frac {6}{q^{3/2}}}+{\frac {4}{q^{5/2}}}-{\frac {7}{q^{7/2}}}+{\frac {3}{q^{9/2}}}-{\frac {4}{q^{11/2}}}+{\frac {1}{q^{13/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$-za^{7}+a^{7}z^{-3}+z^{5}a^{5}+4z^{3}a^{5}+3za^{5}-2a^{5}z^{-1}-3a^{5}z^{-3}-z^{7}a^{3}-5z^{5}a^{3}-7z^{3}a^{3}-2za^{3}+4a^{3}z^{-1}+3a^{3}z^{-3}+z^{5}a+3z^{3}a-2az^{-1}-az^{-3}$ (db)
Kauffman polynomial	$-z^{2}a^{8}-4z^{3}a^{7}+3za^{7}+2a^{7}z^{-1}-a^{7}z^{-3}-2z^{6}a^{6}+4z^{4}a^{6}-3z^{2}a^{6}+3a^{6}z^{-2}-4a^{6}-4z^{7}a^{5}+16z^{5}a^{5}-24z^{3}a^{5}+11za^{5}+3a^{5}z^{-1}-3a^{5}z^{-3}-2z^{8}a^{4}+4z^{6}a^{4}+3z^{4}a^{4}-4z^{2}a^{4}+6a^{4}z^{-2}-7a^{4}-7z^{7}a^{3}+28z^{5}a^{3}-32z^{3}a^{3}+11za^{3}+3a^{3}z^{-1}-3a^{3}z^{-3}-2z^{8}a^{2}+5z^{6}a^{2}+2z^{4}a^{2}-3z^{2}a^{2}+3a^{2}z^{-2}-4a^{2}-3z^{7}a+12z^{5}a-12z^{3}a+3za+2az^{-1}-az^{-3}-z^{6}+3z^{4}-z^{2}$ (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		\

-5

-4

-3

-2

-1

0

1

2

3

χ

4

1

-1

2

0

1

0

-2

5

2

3

-4

1

2

3

2

-6

6

3

-8

1

2

5

4

-10

3

2

1

-12

3

-14

1

-1

Integral Khovanov Homology

(db, data source)

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