L10n78

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n78 at Knotilus!
	[edit L10n78 Quick Notes]

[edit L10n78 Further Notes and Views]

Link Presentations

[edit Notes on L10n78's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_5,14,6,15 X₃₈₄₉ X_2,16,3,15 X_16,7,17,8 X_9,18,10,19 X_11,20,12,13 X_13,12,14,5 X_17,1,18,4 X_19,10,20,11
Gauss code	{1, -4, -3, 9}, {-2, -1, 5, 3, -6, 10, -7, 8}, {-8, 2, 4, -5, -9, 6, -10, 7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(3)^{2}t(2)^{3}-t(1)t(3)^{3}t(2)^{2}+t(1)t(3)^{2}t(2)^{2}-t(3)^{2}t(2)^{2}+t(3)t(2)^{2}-t(1)t(3)^{2}t(2)+t(1)t(3)t(2)-t(3)t(2)+t(2)-t(1)t(3)}{{\sqrt {t(1)}}t(2)^{3/2}t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{-2}-q^{-3}+3q^{-4}-3q^{-5}+4q^{-6}-3q^{-7}+2q^{-8}-q^{-9}+q^{-10}$ (db)
Signature	-4 (db)
HOMFLY-PT polynomial	$a^{8}z^{4}+4a^{8}z^{2}+a^{8}z^{-2}+4a^{8}-a^{6}z^{6}-6a^{6}z^{4}-13a^{6}z^{2}-2a^{6}z^{-2}-11a^{6}+2a^{4}z^{4}+8a^{4}z^{2}+a^{4}z^{-2}+7a^{4}$ (db)
Kauffman polynomial	$z^{4}a^{12}-3z^{2}a^{12}+a^{12}+z^{5}a^{11}-2z^{3}a^{11}+z^{6}a^{10}-2z^{4}a^{10}+z^{2}a^{10}+z^{7}a^{9}-3z^{5}a^{9}+4z^{3}a^{9}+z^{8}a^{8}-5z^{6}a^{8}+11z^{4}a^{8}-8z^{2}a^{8}-a^{8}z^{-2}+5a^{8}+2z^{7}a^{7}-9z^{5}a^{7}+17z^{3}a^{7}-11za^{7}+2a^{7}z^{-1}+z^{8}a^{6}-6z^{6}a^{6}+17z^{4}a^{6}-23z^{2}a^{6}-2a^{6}z^{-2}+13a^{6}+z^{7}a^{5}-5z^{5}a^{5}+11z^{3}a^{5}-11za^{5}+2a^{5}z^{-1}+3z^{4}a^{4}-11z^{2}a^{4}-a^{4}z^{-2}+8a^{4}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-3

2

-5

1

2

1

-7

2

-9

1

0

-11

3

2

1

-13

1

2

1

-15

1

2

-1

-17

1

-19

1

0

-21

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-5$	$i=-3$
$r=-8$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-4$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{3}$
$r=-3$	${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-2$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$	${\mathbb {Z} }^{2}$
$r=-1$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=0$	${\mathbb {Z} }^{2}$	${\mathbb {Z} }^{2}$