L10n59

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n59 at Knotilus!
	[edit L10n59 Quick Notes]

[edit L10n59 Further Notes and Views]

Link Presentations

[edit Notes on L10n59's Link Presentations]

Planar diagram presentation	X_12,1,13,2 X_7,17,8,16 X_5,1,6,10 X₃₇₄₆ X_9,5,10,4 X_20,17,11,18 X_18,13,19,14 X_14,19,15,20 X_2,11,3,12 X_15,9,16,8
Gauss code	{1, -9, -4, 5, -3, 4, -2, 10, -5, 3}, {9, -1, 7, -8, -10, 2, 6, -7, 8, -6}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)(t(2)-t(1))(t(1)t(2)-1)}{t(1)^{3/2}t(2)^{3/2}}}$ (db)
Jones polynomial	$q^{9/2}-q^{7/2}+q^{5/2}-q^{3/2}-{\sqrt {q}}-{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}+{\frac {1}{q^{5/2}}}-{\frac {1}{q^{7/2}}}+{\frac {1}{q^{9/2}}}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$az^{5}-z^{5}a^{-1}-a^{3}z^{3}+6az^{3}-6z^{3}a^{-1}+z^{3}a^{-3}-3a^{3}z+11az-11za^{-1}+3za^{-3}-2a^{3}z^{-1}+7az^{-1}-7a^{-1}z^{-1}+2a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-a^{3}z^{7}-az^{7}-z^{7}a^{-1}-z^{7}a^{-3}-a^{4}z^{6}-2a^{2}z^{6}-2z^{6}a^{-2}-z^{6}a^{-4}-2z^{6}+5a^{3}z^{5}+7az^{5}+7z^{5}a^{-1}+5z^{5}a^{-3}+5a^{4}z^{4}+12a^{2}z^{4}+12z^{4}a^{-2}+5z^{4}a^{-4}+14z^{4}-6a^{3}z^{3}-14az^{3}-14z^{3}a^{-1}-6z^{3}a^{-3}-6a^{4}z^{2}-18a^{2}z^{2}-18z^{2}a^{-2}-6z^{2}a^{-4}-24z^{2}+4a^{3}z+14az+14za^{-1}+4za^{-3}+2a^{4}+8a^{2}+8a^{-2}+2a^{-4}+13-2a^{3}z^{-1}-7az^{-1}-7a^{-1}z^{-1}-2a^{-3}z^{-1}$ (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

4

5

χ

10

1

-1

8

0

6

1

0

4

1

0

2

1

2

0

1

4

1

2

-2

1

2

-4

1

0

-6

1

0

-8

0

-10

1

-1

Integral Khovanov Homology

(db, data source)

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