L10n67

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n67 at Knotilus!
	[edit L10n67 Quick Notes]

[edit L10n67 Further Notes and Views]

Link Presentations

[edit Notes on L10n67's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_10,3,11,4 X_11,16,12,17 X_13,19,14,18 X_17,20,18,9 X_19,13,20,12 X_8,16,5,15 X_14,8,15,7 X₂₅₃₆ X_4,9,1,10
Gauss code	{1, -9, 2, -10}, {9, -1, 8, -7}, {10, -2, -3, 6, -4, -8, 7, 3, -5, 4, -6, 5}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {t(2)t(3)^{3}-2t(1)t(3)^{2}+t(1)t(2)t(3)^{2}-4t(2)t(3)^{2}+t(3)^{2}+4t(1)t(3)-t(1)t(2)t(3)+2t(2)t(3)-t(3)-t(1)}{{\sqrt {t(1)}}{\sqrt {t(2)}}t(3)^{3/2}}}$ (db)
Jones polynomial	$2q^{2}-3q+6-6q^{-1}+6q^{-2}-5q^{-3}+5q^{-4}-2q^{-5}+q^{-6}$ (db)
Signature	0 (db)
HOMFLY-PT polynomial	$a^{6}z^{-2}+a^{6}-3z^{2}a^{4}-3a^{4}z^{-2}-5a^{4}+2z^{4}a^{2}+6z^{2}a^{2}+4a^{2}z^{-2}+8a^{2}-4z^{2}-3z^{-2}-6+a^{-2}z^{-2}+2a^{-2}$ (db)
Kauffman polynomial	$a^{6}z^{6}-4a^{6}z^{4}+6a^{6}z^{2}+a^{6}z^{-2}-4a^{6}+2a^{5}z^{7}-6a^{5}z^{5}+3a^{5}z^{3}+a^{5}z-a^{5}z^{-1}+a^{4}z^{8}+3a^{4}z^{6}-21a^{4}z^{4}+25a^{4}z^{2}+3a^{4}z^{-2}-14a^{4}+6a^{3}z^{7}-18a^{3}z^{5}+11a^{3}z^{3}+a^{3}z-a^{3}z^{-1}+a^{2}z^{8}+6a^{2}z^{6}-30a^{2}z^{4}+39a^{2}z^{2}+3z^{2}a^{-2}+4a^{2}z^{-2}+a^{-2}z^{-2}-21a^{2}-4a^{-2}+4az^{7}-11az^{5}+z^{5}a^{-1}+9az^{3}+z^{3}a^{-1}+az+za^{-1}-az^{-1}-a^{-1}z^{-1}+4z^{6}-13z^{4}+23z^{2}+3z^{-2}-14$ (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

5

2

3

1

-1

1

5

2

3

-1

4

0

-3

2

3

1

0

-5

3

4

1

-7

2

0

-9

1

4

3

-11

1

-1

-13

1

Integral Khovanov Homology

(db, data source)

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