L10n112

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n112 at Knotilus!
	[edit L10n112 Quick Notes]

[edit L10n112 Further Notes and Views]

Link Presentations

[edit Notes on L10n112's Link Presentations]

Planar diagram presentation	X₆₁₇₂ X₂₅₃₆ X_11,19,12,18 X_3,11,4,10 X_9,1,10,4 X_7,15,8,14 X_13,5,14,8 X_15,17,16,20 X_19,13,20,16 X_17,9,18,12
Gauss code	{1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, -3, 10}, {-7, 6, -8, 9}, {-10, 3, -9, 8}

A Braid Representative

A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	${\frac {-t(1)t(2)+t(1)t(3)t(2)-t(3)t(2)+t(1)t(4)t(2)-t(1)t(3)t(4)t(2)+t(3)t(4)t(2)+2t(1)t(5)t(2)-t(1)t(3)t(5)t(2)+t(3)t(5)t(2)-t(1)t(4)t(5)t(2)-t(5)t(2)+t(3)-t(1)t(4)+t(1)t(3)t(4)-2t(3)t(4)+t(4)-t(1)t(5)-t(3)t(5)+t(1)t(4)t(5)+t(3)t(4)t(5)-t(4)t(5)+t(5)}{{\sqrt {t(1)}}{\sqrt {t(2)}}{\sqrt {t(3)}}{\sqrt {t(4)}}{\sqrt {t(5)}}}}$ (db)
Jones polynomial	$q^{9}-q^{8}+6q^{7}-5q^{6}+11q^{5}-5q^{4}+10q^{3}-5q^{2}+4q$ (db)
Signature	2 (db)
HOMFLY-PT polynomial	$a^{-10}z^{-4}+a^{-10}z^{-2}-4a^{-8}z^{-4}-7a^{-8}z^{-2}-4a^{-8}+6a^{-6}z^{-4}+6z^{2}a^{-6}+15a^{-6}z^{-2}+14a^{-6}-3z^{4}a^{-4}-4a^{-4}z^{-4}-10z^{2}a^{-4}-13a^{-4}z^{-2}-16a^{-4}+a^{-2}z^{-4}+4z^{2}a^{-2}+4a^{-2}z^{-2}+6a^{-2}$ (db)
Kauffman polynomial	$z^{8}a^{-6}+z^{8}a^{-8}+5z^{7}a^{-5}+6z^{7}a^{-7}+z^{7}a^{-9}+10z^{6}a^{-4}+12z^{6}a^{-6}+3z^{6}a^{-8}+z^{6}a^{-10}+6z^{5}a^{-3}-6z^{5}a^{-7}-25z^{4}a^{-4}-39z^{4}a^{-6}-19z^{4}a^{-8}-5z^{4}a^{-10}-10z^{3}a^{-3}-30z^{3}a^{-5}-30z^{3}a^{-7}-10z^{3}a^{-9}+10z^{2}a^{-2}+30z^{2}a^{-4}+40z^{2}a^{-6}+30z^{2}a^{-8}+10z^{2}a^{-10}+20za^{-3}+55za^{-5}+55za^{-7}+20za^{-9}-10a^{-2}-25a^{-4}-31a^{-6}-25a^{-8}-10a^{-10}-15a^{-3}z^{-1}-41a^{-5}z^{-1}-41a^{-7}z^{-1}-15a^{-9}z^{-1}+5a^{-2}z^{-2}+14a^{-4}z^{-2}+18a^{-6}z^{-2}+14a^{-8}z^{-2}+5a^{-10}z^{-2}+4a^{-3}z^{-3}+12a^{-5}z^{-3}+12a^{-7}z^{-3}+4a^{-9}z^{-3}-a^{-2}z^{-4}-4a^{-4}z^{-4}-6a^{-6}z^{-4}-4a^{-8}z^{-4}-a^{-10}z^{-4}$ (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		\

0

1

2

3

4

5

6

7

8

χ

19

1

17

1

0

15

5

13

1

11

5

6

9

4

10

6

7

6

1

5

1

4

5

3

5

6

-1

1

4

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=1$	$i=3$	$i=5$
$r=0$	${\mathbb {Z} }^{4}$	${\mathbb {Z} }^{5}$	${\mathbb {Z} }$
$r=1$	${\mathbb {Z} }^{6}$
$r=2$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$	${\mathbb {Z} }^{6}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=4$	${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }^{11}$
$r=5$	${\mathbb {Z} }^{5}$
$r=6$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{5}$	${\mathbb {Z} }^{5}$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=8$	${\mathbb {Z} }$	${\mathbb {Z} }$