L10n8

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n8 at Knotilus!
	[edit L10n8 Quick Notes]

[edit L10n8 Further Notes and Views]

Link Presentations

[edit Notes on L10n8's Link Presentations]

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

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {2(u-1)(v-1)}{{\sqrt {u}}{\sqrt {v}}}}$ (db)
Jones polynomial	$-q^{9/2}+q^{7/2}-{\frac {1}{q^{7/2}}}-2q^{5/2}+{\frac {1}{q^{5/2}}}+3q^{3/2}-{\frac {2}{q^{3/2}}}-3{\sqrt {q}}+{\frac {2}{\sqrt {q}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$za^{3}+a^{3}z^{-1}-z^{3}a-2za-az^{-1}-z^{3}a^{-1}-za^{-1}-a^{-1}z^{-1}+2za^{-3}+2a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$z^{3}a^{-5}-3za^{-5}+a^{-5}z^{-1}+z^{4}a^{-4}-2z^{2}a^{-4}+a^{-4}+z^{7}a^{-3}+a^{3}z^{5}-6z^{5}a^{-3}-4a^{3}z^{3}+14z^{3}a^{-3}+3a^{3}z-11za^{-3}-a^{3}z^{-1}+2a^{-3}z^{-1}+z^{8}a^{-2}+a^{2}z^{6}-6z^{6}a^{-2}-3a^{2}z^{4}+13z^{4}a^{-2}-9z^{2}a^{-2}+a^{2}+3a^{-2}+az^{7}+2z^{7}a^{-1}-3az^{5}-10z^{5}a^{-1}+17z^{3}a^{-1}+3az-8za^{-1}-az^{-1}+a^{-1}z^{-1}+z^{8}-5z^{6}+9z^{4}-7z^{2}+2$ (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

10

1

8

0

6

2

1

4

1

-1

2

0

2

3

1

-2

1

0

-4

1

2

1

-6

0

-8

1

Integral Khovanov Homology

(db, data source)

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