L11a171

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_20,9,21,10 X_14,5,15,6 X_18,12,19,11 X_10,4,11,3 X_12,7,13,8 X_16,13,17,14 X_22,17,7,18 X_6,15,1,16 X_4,21,5,22 X_2,20,3,19
Gauss code	{1, -11, 5, -10, 3, -9}, {6, -1, 2, -5, 4, -6, 7, -3, 9, -7, 8, -4, 11, -2, 10, -8}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-2 v 2 u 4 + 3 v u 4 - u 4 + 7 v 2 u 3 -12 v u 3 + 5 u 3 -9 v 2 u 2 + 17 v u 2 -9 u 2 + 5 v 2 u -12 v u + 7 u - v 2 + 3 v -2$ (db)
Jones polynomial	$q^{3/2}-5 \sqrt{q}+\frac{11}{\sqrt{q}}-\frac{20}{q^{3/2}}+\frac{26}{q^{5/2}}-\frac{31}{q^{7/2}}+\frac{31}{q^{9/2}}-\frac{27}{q^{11/2}}+\frac{20}{q^{13/2}}-\frac{12}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z 5 a 7 + z 3 a 7 - z 7 a 5 -2 z 5 a 5 - z 3 a 5 - z 7 a 3 -2 z 5 a 3 - z 3 a 3 + a 3 z -1 + z 5 a + z 3 a - z a - a z -1$ (db)
Kauffman polynomial	$- z 5 a 11 -5 z 6 a 10 + 3 z 4 a 10 -12 z 7 a 9 + 14 z 5 a 9 -5 z 3 a 9 -17 z 8 a 8 + 25 z 6 a 8 -13 z 4 a 8 + 2 z 2 a 8 -13 z 9 a 7 + 7 z 7 a 7 + 14 z 5 a 7 -8 z 3 a 7 -4 z 10 a 6 -26 z 8 a 6 + 64 z 6 a 6 -38 z 4 a 6 + 6 z 2 a 6 -23 z 9 a 5 + 34 z 7 a 5 - z 5 a 5 -6 z 3 a 5 -4 z 10 a 4 -19 z 8 a 4 + 54 z 6 a 4 -34 z 4 a 4 + 6 z 2 a 4 -10 z 9 a 3 + 10 z 7 a 3 + 9 z 5 a 3 -8 z 3 a 3 - z a 3 + a 3 z -1 -10 z 8 a 2 + 19 z 6 a 2 -11 z 4 a 2 + 2 z 2 a 2 - a 2 -5 z 7 a + 9 z 5 a -5 z 3 a - z a + a z -1 - z 6 + z 4$ (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 L11a171. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -4$	$i = -2$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = -5$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = -4$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -3$	${\mathbb Z}^{16}\oplus{\mathbb Z}_2^{15}$	${\mathbb Z}^{15}$
$r = -2$	${\mathbb Z}^{15}\oplus{\mathbb Z}_2^{16}$	${\mathbb Z}^{17}$
$r = -1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 0$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{13}$
$r = 1$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$