L10n39

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n39 at Knotilus!
	[edit L10n39 Quick Notes]

[edit L10n39 Further Notes and Views]

Link Presentations

[edit Notes on L10n39's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_12,4,13,3 X_14,10,15,9 X_16,12,17,11 X_10,16,11,15 X_17,5,18,20 X_7,19,8,18 X_19,9,20,8 X₂₅₃₆ X_4,14,1,13
Gauss code	{1, -9, 2, -10}, {9, -1, -7, 8, 3, -5, 4, -2, 10, -3, 5, -4, -6, 7, -8, 6}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-{\frac {(t(1)-1)(t(2)-1)\left(t(2)^{2}-t(2)+1\right)\left(t(2)^{2}+t(2)+1\right)}{{\sqrt {t(1)}}t(2)^{5/2}}}$ (db)
Jones polynomial	$-4q^{9/2}+3q^{7/2}-4q^{5/2}+q^{3/2}-q^{17/2}+3q^{15/2}-3q^{13/2}+4q^{11/2}-{\sqrt {q}}$ (db)
Signature	5 (db)
HOMFLY-PT polynomial	$-za^{-9}-a^{-9}z^{-1}+z^{5}a^{-7}+5z^{3}a^{-7}+8za^{-7}+5a^{-7}z^{-1}-z^{7}a^{-5}-6z^{5}a^{-5}-13z^{3}a^{-5}-15za^{-5}-8a^{-5}z^{-1}+z^{5}a^{-3}+5z^{3}a^{-3}+8za^{-3}+4a^{-3}z^{-1}$ (db)
Kauffman polynomial	$-z^{8}a^{-4}-z^{8}a^{-6}-z^{7}a^{-3}-5z^{7}a^{-5}-4z^{7}a^{-7}+3z^{6}a^{-4}-2z^{6}a^{-6}-5z^{6}a^{-8}+6z^{5}a^{-3}+24z^{5}a^{-5}+16z^{5}a^{-7}-2z^{5}a^{-9}+3z^{4}a^{-4}+23z^{4}a^{-6}+20z^{4}a^{-8}-13z^{3}a^{-3}-35z^{3}a^{-5}-18z^{3}a^{-7}+4z^{3}a^{-9}-13z^{2}a^{-4}-33z^{2}a^{-6}-23z^{2}a^{-8}-3z^{2}a^{-10}+12za^{-3}+23za^{-5}+11za^{-7}-za^{-9}-za^{-11}+8a^{-4}+14a^{-6}+9a^{-8}+2a^{-10}-4a^{-3}z^{-1}-8a^{-5}z^{-1}-5a^{-7}z^{-1}-a^{-9}z^{-1}$ (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

18

1

16

2

-2

14

2

0

12

3

2

-1

10

1

2

1

0

8

2

3

1

6

2

1

4

1

4

3

2

0

1

Integral Khovanov Homology

(db, data source)

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