L10n95

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n95 at Knotilus!
	[edit L10n95 Quick Notes]

[edit L10n95 Further Notes and Views]

Link Presentations

[edit Notes on L10n95's Link Presentations]

Planar diagram presentation	X₈₁₉₂ X_18,12,19,11 X_3,10,4,11 X_19,2,20,3 X_16,8,17,7 X_9,13,10,20 X_12,18,7,17 X_4,13,5,14 X_14,5,15,6 X_6,15,1,16
Gauss code	{1, 4, -3, -8, 9, -10}, {5, -1, -6, 3, 2, -7}, {8, -9, 10, -5, 7, -2, -4, 6}

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

Polynomial invariants

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

3

2

1

0

-1

4

1

3

-3

3

0

-5

3

2

1

-7

2

3

1

-9

2

3

-1

-11

2

-13

1

2

-1

-15

1

Integral Khovanov Homology

(db, data source)

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