L11a139

(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_12,5,13,6 X_16,12,17,11 X_6,18,1,17 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 $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - u 4 + 5 v 2 u 3 -10 v u 3 + 5 u 3 -8 v 2 u 2 + 15 v u 2 -8 u 2 + 5 v 2 u -10 v u + 5 u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{13/2}+4 q^{11/2}-9 q^{9/2}+16 q^{7/2}-22 q^{5/2}+25 q^{3/2}-26 \sqrt{q}+\frac{22}{\sqrt{q}}-\frac{17}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -3 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 3 a z 3 -5 z 3 a -1 + 4 z 3 a -3 - z 3 a -5 -3 z a -1 + 3 z a -3 - z a -5 + a 3 z -1 - a z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -7 a z 9 -14 z 9 a -1 -7 z 9 a -3 -9 a 2 z 8 -21 z 8 a -2 -10 z 8 a -4 -20 z 8 -5 a 3 z 7 + 2 a z 7 + 11 z 7 a -1 -4 z 7 a -3 -8 z 7 a -5 - a 4 z 6 + 19 a 2 z 6 + 49 z 6 a -2 + 12 z 6 a -4 -4 z 6 a -6 + 53 z 6 + 10 a 3 z 5 + 22 a z 5 + 25 z 5 a -1 + 25 z 5 a -3 + 11 z 5 a -5 - z 5 a -7 + a 4 z 4 -11 a 2 z 4 -34 z 4 a -2 -5 z 4 a -4 + 5 z 4 a -6 -36 z 4 -5 a 3 z 3 -17 a z 3 -26 z 3 a -1 -22 z 3 a -3 -7 z 3 a -5 + z 3 a -7 + 2 a 2 z 2 + 8 z 2 a -2 -2 z 2 a -6 + 8 z 2 - a 3 z + a z + 6 z a -1 + 6 z a -3 + 2 z a -5 - a 2 + a 3 z -1 + a z -1$ (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 L11a139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{7}$
$r = -1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 2$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$