L11n119

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_3,15,4,14 X_16,10,17,9 X_11,21,12,20 X_21,9,22,8 X_7,19,8,18 X_19,13,20,12 X_10,16,11,15 X_17,5,18,22 X₂₅₃₆ X_13,1,14,4
Gauss code	{1, -10, -2, 11}, {10, -1, -6, 5, 3, -8, -4, 7, -11, 2, 8, -3, -9, 6, -7, 4, -5, 9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v u 3 + 2 u 3 + 3 v u 2 -3 u 2 -3 v u + 3 u + 2 v -2$ (db)
Jones polynomial	$2 q 17 / 2 -4 q 15 / 2 + 6 q 13 / 2 -7 q 11 / 2 + 7 q 9 / 2 -7 q 7 / 2 + 4 q 5 / 2 -3 q 3 / 2$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$-2 z 5 a -5 + 3 z 3 a -3 -8 z 3 a -5 + 2 z 3 a -7 + 7 z a -3 -11 z a -5 + 4 z a -7 + 3 a -3 z -1 -5 a -5 z -1 + 2 a -7 z -1$ (db)
Kauffman polynomial	$- z 8 a -6 - z 8 a -8 -4 z 7 a -5 -5 z 7 a -7 - z 7 a -9 -3 z 6 a -4 -4 z 6 a -6 - z 6 a -8 + 12 z 5 a -5 + 11 z 5 a -7 - z 5 a -9 + 6 z 4 a -4 + 12 z 4 a -6 + 3 z 4 a -8 -3 z 4 a -10 -6 z 3 a -3 -22 z 3 a -5 -12 z 3 a -7 + 4 z 3 a -9 -9 z 2 a -4 -14 z 2 a -6 + 5 z 2 a -10 + 10 z a -3 + 17 z a -5 + 7 z a -7 + 5 a -4 + 5 a -6 - a -10 -3 a -3 z -1 -5 a -5 z -1 -2 a -7 z -1$ (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 =

3 is the signature of L11n119. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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