L10n99

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_7,15,8,14 X_13,5,14,8 X_11,18,12,19 X_15,20,16,17 X_19,16,20,9 X_17,12,18,13 X₂₅₃₆ X_9,1,10,4
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 $u$ , $v$ , $w$ , ...)	$-{\frac {(x-1)\left(uvwx-uvw-2uvx-uwx+ux-vwx+vx+2wx+x^{2}-x\right)}{{\sqrt {u}}{\sqrt {v}}{\sqrt {w}}x^{3/2}}}$ (db)
Jones polynomial	$-q^{7/2}+2q^{5/2}-6q^{3/2}+7{\sqrt {q}}-{\frac {9}{\sqrt {q}}}+{\frac {7}{q^{3/2}}}-{\frac {9}{q^{5/2}}}+{\frac {4}{q^{7/2}}}-{\frac {3}{q^{9/2}}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a^{5}z^{-3}+2a^{5}z^{-1}+a^{3}z^{3}-3a^{3}z^{-3}-3a^{3}z-6a^{3}z^{-1}-za^{-3}-a^{-3}z^{-1}-az^{5}-az^{3}+3az^{-3}+2z^{3}a^{-1}-a^{-1}z^{-3}+2az+5az^{-1}+2za^{-1}$ (db)
Kauffman polynomial	$6a^{5}z^{3}-a^{5}z^{-3}-10a^{5}z+6a^{5}z^{-1}+3a^{4}z^{6}-6a^{4}z^{4}+15a^{4}z^{2}+3a^{4}z^{-2}-13a^{4}+4a^{3}z^{7}-9a^{3}z^{5}+z^{5}a^{-3}+19a^{3}z^{3}-3z^{3}a^{-3}-3a^{3}z^{-3}-23a^{3}z+3za^{-3}+14a^{3}z^{-1}-a^{-3}z^{-1}+a^{2}z^{8}+8a^{2}z^{6}+2z^{6}a^{-2}-24a^{2}z^{4}-3z^{4}a^{-2}+33a^{2}z^{2}+6a^{2}z^{-2}-24a^{2}+a^{-2}+7az^{7}+3z^{7}a^{-1}-15az^{5}-5z^{5}a^{-1}+19az^{3}+3z^{3}a^{-1}-3az^{-3}-a^{-1}z^{-3}-20az-4za^{-1}+12az^{-1}+3a^{-1}z^{-1}+z^{8}+7z^{6}-21z^{4}+18z^{2}+3z^{-2}-11$ (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

3

4

χ

8

1

6

2

1

-1

4

2

3

2

-1

0

6

4

2

-2

5

7

2

-4

4

2

-6

5

-8

3

4

-1

-10

3

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-2$	$i=0$
$r=-4$	${\mathbb {Z} }^{3}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{4}$
$r=-2$	${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-1$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=0$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{6}$
$r=1$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=2$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=3$	${\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=4$	${\mathbb {Z} }$	${\mathbb {Z} }$