L11n151

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_18,9,19,10 X₆₇₁₈ X_22,19,7,20 X_12,5,13,6 X_3,10,4,11 X_4,15,5,16 X_11,16,12,17 X_20,13,21,14 X_14,21,15,22 X_17,2,18,3
Gauss code	{1, 11, -6, -7, 5, -3}, {3, -1, 2, 6, -8, -5, 9, -10, 7, 8, -11, -2, 4, -9, 10, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 2 v 2 u 3 - v u 3 - v 2 u 2 + v u 2 - u 2 - v u + 2 u -2$ (db)
Jones polynomial	$-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{3}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}}$ (db)
Signature	-7 (db)
HOMFLY-PT polynomial	$- z a 13 - a 13 z -1 + z 5 a 11 + 5 z 3 a 11 + 6 z a 11 + 2 a 11 z -1 - z 7 a 9 -5 z 5 a 9 -6 z 3 a 9 - z a 9 - z 7 a 7 -6 z 5 a 7 -11 z 3 a 7 -7 z a 7 - a 7 z -1$ (db)
Kauffman polynomial	$- z 2 a 16 -2 z 3 a 15 + z a 15 -3 z 4 a 14 + 4 z 2 a 14 -2 a 14 - z 7 a 13 + 2 z 5 a 13 - z 3 a 13 - z a 13 + a 13 z -1 -2 z 8 a 12 + 10 z 6 a 12 -20 z 4 a 12 + 19 z 2 a 12 -5 a 12 - z 9 a 11 + 4 z 7 a 11 -7 z 5 a 11 + 12 z 3 a 11 -9 z a 11 + 2 a 11 z -1 -3 z 8 a 10 + 14 z 6 a 10 -19 z 4 a 10 + 11 z 2 a 10 -3 a 10 - z 9 a 9 + 4 z 7 a 9 -3 z 5 a 9 - z 8 a 8 + 4 z 6 a 8 -2 z 4 a 8 -3 z 2 a 8 + a 8 - z 7 a 7 + 6 z 5 a 7 -11 z 3 a 7 + 7 z a 7 - a 7 z -1$ (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 =

-7 is the signature of L11n151. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -8$	$i = -6$
$r = -9$	${\mathbb Z}$
$r = -8$	${\mathbb Z}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{3}$
$r = -5$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -4$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -3$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -2$	${\mathbb Z}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -1$	${\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}$	${\mathbb Z}$