L10n6

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n6 at Knotilus!
	[edit L10n6 Quick Notes]

[edit L10n6 Further Notes and Views]

Link Presentations

[edit Notes on L10n6's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_16,7,17,8 X_4,17,1,18 X_9,14,10,15 X₈₄₉₃ X_5,11,6,10 X_13,5,14,20 X_11,19,12,18 X_19,13,20,12 X_2,16,3,15
Gauss code	{1, -10, 5, -3}, {-6, -1, 2, -5, -4, 6, -8, 9, -7, 4, 10, -2, 3, 8, -9, 7}

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

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	$-q^{13/2}+2q^{11/2}-3q^{9/2}+4q^{7/2}-4q^{5/2}+4q^{3/2}-4{\sqrt {q}}+{\frac {1}{\sqrt {q}}}-{\frac {1}{q^{3/2}}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z^{5}a^{-3}-2z^{3}a^{-1}+4z^{3}a^{-3}-z^{3}a^{-5}+az-5za^{-1}+6za^{-3}-2za^{-5}+2az^{-1}-4a^{-1}z^{-1}+3a^{-3}z^{-1}-a^{-5}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-2}-z^{8}a^{-4}-z^{7}a^{-1}-3z^{7}a^{-3}-2z^{7}a^{-5}+4z^{6}a^{-2}+2z^{6}a^{-4}-2z^{6}a^{-6}+5z^{5}a^{-1}+12z^{5}a^{-3}+6z^{5}a^{-5}-z^{5}a^{-7}-6z^{4}a^{-2}+z^{4}a^{-4}+6z^{4}a^{-6}-z^{4}-az^{3}-13z^{3}a^{-1}-19z^{3}a^{-3}-4z^{3}a^{-5}+3z^{3}a^{-7}-2z^{2}a^{-2}-5z^{2}a^{-4}-3z^{2}a^{-6}+3az+11za^{-1}+11za^{-3}+2za^{-5}-za^{-7}+3a^{-2}+3a^{-4}+a^{-6}+2-2az^{-1}-4a^{-1}z^{-1}-3a^{-3}z^{-1}-a^{-5}z^{-1}$ (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		\

-2

-1

0

1

2

3

4

5

6

χ

14

1

12

1

-1

10

2

1

8

2

1

-1

6

2

0

4

2

0

2

0

1

4

3

-2

0

-4

1

Integral Khovanov Homology

(db, data source)

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