L10n16

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n16 at Knotilus!
	[edit L10n16 Quick Notes]

[edit L10n16 Further Notes and Views]

Link Presentations

[edit Notes on L10n16's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X_18,7,19,8 X_4,19,1,20 X_9,14,10,15 X₈₄₉₃ X_12,5,13,6 X_20,13,5,14 X_11,16,12,17 X_15,10,16,11 X_2,18,3,17
Gauss code	{1, -10, 5, -3}, {6, -1, 2, -5, -4, 9, -8, -6, 7, 4, -9, 8, 10, -2, 3, -7}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {(t(1)-1)(t(2)-1)\left(2t(2)^{2}-3t(2)+2\right)}{{\sqrt {t(1)}}t(2)^{3/2}}}$ (db)
Jones polynomial	${\frac {9}{q^{9/2}}}-{\frac {9}{q^{7/2}}}+{\frac {5}{q^{5/2}}}-{\frac {3}{q^{3/2}}}-{\frac {1}{q^{19/2}}}+{\frac {4}{q^{17/2}}}-{\frac {6}{q^{15/2}}}+{\frac {9}{q^{13/2}}}-{\frac {10}{q^{11/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a^{9}z-3a^{7}z^{3}-5a^{7}z-2a^{7}z^{-1}+2a^{5}z^{5}+7a^{5}z^{3}+10a^{5}z+5a^{5}z^{-1}-3a^{3}z^{3}-6a^{3}z-3a^{3}z^{-1}$ (db)
Kauffman polynomial	$-z^{5}a^{11}+z^{3}a^{11}-4z^{6}a^{10}+8z^{4}a^{10}-3z^{2}a^{10}-a^{10}-5z^{7}a^{9}+9z^{5}a^{9}-3z^{3}a^{9}+2za^{9}-2z^{8}a^{8}-6z^{6}a^{8}+20z^{4}a^{8}-10z^{2}a^{8}-10z^{7}a^{7}+21z^{5}a^{7}-17z^{3}a^{7}+9za^{7}-2a^{7}z^{-1}-2z^{8}a^{6}-5z^{6}a^{6}+15z^{4}a^{6}-14z^{2}a^{6}+5a^{6}-5z^{7}a^{5}+11z^{5}a^{5}-19z^{3}a^{5}+15za^{5}-5a^{5}z^{-1}-3z^{6}a^{4}+3z^{4}a^{4}-7z^{2}a^{4}+5a^{4}-6z^{3}a^{3}+8za^{3}-3a^{3}z^{-1}$ (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		\

-8

-7

-6

-5

-4

-3

-2

-1

0

χ

-2

3

-4

4

2

-2

-6

5

1

4

-8

4

0

-10

6

5

1

-12

3

4

1

-14

3

6

-3

-16

1

3

2

-18

3

-3

-20

1

Integral Khovanov Homology

(db, data source)

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