L10n112

(Knotscape image)

See the full Thistlethwaite Link Table (up to 11 crossings).

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

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

The coefficients of the monomials

t r q j

are shown, along with their alternating sums

χ

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j -2 r = s + 1

j -2 r = s -1

, where

s =

2 is the signature of L10n112. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 1$	$i = 3$	$i = 5$
$r = 0$	${\mathbb Z}^{4}$	${\mathbb Z}^{5}$	${\mathbb Z}$
$r = 1$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2$	${\mathbb Z}^{11}$
$r = 5$	${\mathbb Z}^{5}$
$r = 6$	${\mathbb Z}\oplus{\mathbb Z}_2^{5}$	${\mathbb Z}^{5}$
$r = 7$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 8$	${\mathbb Z}$	${\mathbb Z}$