L10n83

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n83 at Knotilus!
	[edit L10n83 Quick Notes]

[edit L10n83 Further Notes and Views]

Link Presentations

[edit Notes on L10n83's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,7,13,8 X_4,13,1,14 X_5,16,6,17 X₈₄₉₃ X_17,20,18,15 X_19,10,20,11 X_9,18,10,19 X_15,14,16,5 X_2,12,3,11
Gauss code	{1, -10, 5, -3}, {-9, 4, -6, 8, -7, 6}, {-4, -1, 2, -5, -8, 7, 10, -2, 3, 9}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(u-1)(w-1)^{2}(vw+1)}{{\sqrt {u}}{\sqrt {v}}w^{3/2}}}$ (db)
Jones polynomial	$1-2q^{-1}+5q^{-2}-4q^{-3}+6q^{-4}-5q^{-5}+5q^{-6}-3q^{-7}+q^{-8}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$z^{4}a^{6}+2z^{2}a^{6}+a^{6}z^{-2}+a^{6}-z^{6}a^{4}-4z^{4}a^{4}-5z^{2}a^{4}-2a^{4}z^{-2}-4a^{4}+z^{4}a^{2}+3z^{2}a^{2}+a^{2}z^{-2}+3a^{2}$ (db)
Kauffman polynomial	$z^{2}a^{10}-a^{10}+3z^{3}a^{9}-za^{9}+2z^{6}a^{8}-5z^{4}a^{8}+9z^{2}a^{8}-3a^{8}+3z^{7}a^{7}-9z^{5}a^{7}+10z^{3}a^{7}-3za^{7}+z^{8}a^{6}+3z^{6}a^{6}-18z^{4}a^{6}+20z^{2}a^{6}+a^{6}z^{-2}-7a^{6}+5z^{7}a^{5}-15z^{5}a^{5}+9z^{3}a^{5}+za^{5}-2a^{5}z^{-1}+z^{8}a^{4}+2z^{6}a^{4}-17z^{4}a^{4}+18z^{2}a^{4}+2a^{4}z^{-2}-8a^{4}+2z^{7}a^{3}-6z^{5}a^{3}+2z^{3}a^{3}+3za^{3}-2a^{3}z^{-1}+z^{6}a^{2}-4z^{4}a^{2}+6z^{2}a^{2}+a^{2}z^{-2}-4a^{2}$ (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		\

-6

-5

-4

-3

-2

-1

0

1

2

χ

1

-1

1

-1

-3

4

1

3

-5

2

3

1

-7

4

2

-9

1

2

1

-11

4

0

-13

1

3

2

-15

2

-2

-17

1

Integral Khovanov Homology

(db, data source)

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