L11n139

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_11,19,12,18 X_3,10,4,11 X_17,3,18,2 X_5,13,6,12 X₆₇₁₈ X_16,10,17,9 X_20,16,21,15 X_22,14,7,13 X_14,22,15,21 X_19,4,20,5
Gauss code	{1, 4, -3, 11, -5, -6}, {6, -1, 7, 3, -2, 5, 9, -10, 8, -7, -4, 2, -11, -8, 10, -9}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 2 + v u 2 - v 2 u + v u - u + v -1$ (db)
Jones polynomial	$q^{13/2}-q^{11/2}+q^{9/2}-q^{7/2}-\sqrt{q}-\frac{1}{q^{3/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$- z 5 a -3 -5 z 3 a -3 + z 3 a -5 + a z + z a -1 -6 z a -3 + 3 z a -5 + a z -1 -2 a -3 z -1 + a -5 z -1$ (db)
Kauffman polynomial	$- z 9 a -3 - z 9 a -5 - z 8 a -2 -2 z 8 a -4 - z 8 a -6 + 7 z 7 a -3 + 7 z 7 a -5 + 7 z 6 a -2 + 14 z 6 a -4 + 7 z 6 a -6 -15 z 5 a -3 -15 z 5 a -5 -14 z 4 a -2 -29 z 4 a -4 -15 z 4 a -6 - a z 3 + z 3 a -1 + 14 z 3 a -3 + 12 z 3 a -5 + 10 z 2 a -2 + 21 z 2 a -4 + 11 z 2 a -6 + 3 a z - z a -1 -8 z a -3 -4 z a -5 -3 a -2 -5 a -4 -2 a -6 + 1- a z -1 + 2 a -3 z -1 + a -5 z -1$ (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 L11n139. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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