L11n127

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_10,4,11,3 X_22,10,7,9 X₂₇₃₈ X_4,15,5,16 X_5,13,6,12 X_11,16,12,17 X_17,6,18,1 X_14,20,15,19 X_20,14,21,13 X_18,21,19,22
Gauss code	{1, -4, 2, -5, -6, 8}, {4, -1, 3, -2, -7, 6, 10, -9, 5, 7, -8, -11, 9, -10, 11, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + 2 v u 2 - u 2 + v 2 u - v u + u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{5/2}+2 q^{3/2}-3 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{4}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$z a 5 + a 5 z -1 -2 z 3 a 3 -5 z a 3 -2 a 3 z -1 + z 5 a + 4 z 3 a + 5 z a + 2 a z -1 - z 3 a -1 -2 z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$- a 3 z 9 - a z 9 -2 a 4 z 8 -4 a 2 z 8 -2 z 8 - a 5 z 7 + 2 a 3 z 7 + 2 a z 7 - z 7 a -1 + 10 a 4 z 6 + 20 a 2 z 6 + 10 z 6 + 5 a 5 z 5 + 9 a 3 z 5 + 9 a z 5 + 5 z 5 a -1 -13 a 4 z 4 -26 a 2 z 4 -13 z 4 -7 a 5 z 3 -20 a 3 z 3 -20 a z 3 -7 z 3 a -1 + 5 a 4 z 2 + 10 a 2 z 2 + 5 z 2 + 4 a 5 z + 12 a 3 z + 12 a z + 4 z a -1 - a 2 - a 5 z -1 -2 a 3 z -1 -2 a z -1 - a -1 z -1$ (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 =

1 is the signature of L11n127. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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