L10n65

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Link Presentations

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Planar diagram presentation	X₆₁₇₂ X_3,11,4,10 X_11,16,12,17 X_13,19,14,18 X_17,20,18,9 X_19,13,20,12 X_15,8,16,5 X_7,14,8,15 X₂₅₃₆ X_9,1,10,4
Gauss code	{1, -9, -2, 10}, {9, -1, -8, 7}, {-10, 2, -3, 6, -4, 8, -7, 3, -5, 4, -6, 5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {uvw^{3}-3uvw^{2}+3uvw-uv-2uw+u-vw^{3}+2vw^{2}+w^{3}-3w^{2}+3w-1}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$3q^{-5}-5q^{-4}+q^{3}+7q^{-3}-2q^{2}-7q^{-2}+5q+8q^{-1}-6$ (db)
Signature	-2 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-z^{4}a^{4}-3z^{2}a^{4}-3a^{4}z^{-2}-5a^{4}+z^{6}a^{2}+4z^{4}a^{2}+7z^{2}a^{2}+4a^{2}z^{-2}+8a^{2}-2z^{4}-6z^{2}-3z^{-2}-6+z^{2}a^{-2}+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{2}z^{8}+z^{8}+5a^{3}z^{7}+7az^{7}+2z^{7}a^{-1}+7a^{4}z^{6}+10a^{2}z^{6}+z^{6}a^{-2}+4z^{6}+3a^{5}z^{5}-9a^{3}z^{5}-18az^{5}-6z^{5}a^{-1}-18a^{4}z^{4}-37a^{2}z^{4}-4z^{4}a^{-2}-23z^{4}+4a^{3}z^{3}+8az^{3}+4z^{3}a^{-1}+6a^{6}z^{2}+24a^{4}z^{2}+40a^{2}z^{2}+6z^{2}a^{-2}+28z^{2}+a^{5}z+a^{3}z+az+za^{-1}-4a^{6}-14a^{4}-21a^{2}-4a^{-2}-14-a^{5}z^{-1}-a^{3}z^{-1}-az^{-1}-a^{-1}z^{-1}+a^{6}z^{-2}+3a^{4}z^{-2}+4a^{2}z^{-2}+a^{-2}z^{-2}+3z^{-2}$ (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		\

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

1

-1

3

4

1

3

1

2

1

-1

6

4

2

-3

4

5

1

-5

3

0

-7

2

4

2

-9

1

3

-2

-11

3

Integral Khovanov Homology

(db, data source)

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