L10n98

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_13,15,14,20 X_10,3,11,4 X_4,9,1,10 X_16,7,17,8 X_8,15,5,16 X_11,19,12,18 X_19,13,20,12 X_17,9,18,14
Gauss code	{1, -2, 4, -5}, {2, -1, 6, -7}, {5, -4, -8, 9, -3, 10}, {7, -6, -10, 8, -9, 3}

A Braid Representative	{{{braid_table}}}
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(1)t(4)^{2}t(3)^{2}+t(2)t(4)^{2}t(3)^{2}-t(4)^{2}t(3)^{2}-t(2)t(4)t(3)^{2}-t(1)t(4)^{2}t(3)-t(2)t(3)+t(1)t(4)t(3)+t(2)t(4)t(3)+t(1)-t(1)t(2)+t(2)-t(1)t(4)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)t(4)}}$ (db)
Jones polynomial	${\frac {1}{q^{9/2}}}-{\frac {4}{q^{7/2}}}-2q^{5/2}+{\frac {2}{q^{5/2}}}+2q^{3/2}-{\frac {5}{q^{3/2}}}-{\frac {1}{q^{11/2}}}-4{\sqrt {q}}+{\frac {3}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-3}+a^{5}z+3a^{5}z^{-1}-3a^{3}z^{3}-3a^{3}z^{-3}-11a^{3}z-10a^{3}z^{-1}+2az^{5}+10az^{3}+3az^{-3}-2z^{3}a^{-1}-a^{-1}z^{-3}+16az+11az^{-1}-6za^{-1}-4a^{-1}z^{-1}$ (db)
Kauffman polynomial	$-a^{4}z^{8}-a^{2}z^{8}-a^{5}z^{7}-5a^{3}z^{7}-4az^{7}+3a^{4}z^{6}-a^{2}z^{6}-4z^{6}+6a^{5}z^{5}+25a^{3}z^{5}+17az^{5}-2z^{5}a^{-1}+4a^{4}z^{4}+19a^{2}z^{4}-z^{4}a^{-2}+14z^{4}-13a^{5}z^{3}-39a^{3}z^{3}-24az^{3}+2z^{3}a^{-1}-17a^{4}z^{2}-33a^{2}z^{2}-16z^{2}+13a^{5}z+28a^{3}z+21az+3za^{-1}-3za^{-3}+13a^{4}+24a^{2}-a^{-2}+11-6a^{5}z^{-1}-14a^{3}z^{-1}-12az^{-1}-3a^{-1}z^{-1}+a^{-3}z^{-1}-3a^{4}z^{-2}-6a^{2}z^{-2}-3z^{-2}+a^{5}z^{-3}+3a^{3}z^{-3}+3az^{-3}+a^{-1}z^{-3}$ (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

6

2

4

3

0

2

1

2

0

3

4

1

-2

2

1

2

-4

1

4

3

-6

3

1

2

-8

1

4

3

-10

0

-12

1

Integral Khovanov Homology

(db, data source)

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