L11a127

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_14,3,15,4 X_22,10,5,9 X_18,7,19,8 X_8,17,9,18 X_12,19,13,20 X_20,11,21,12 X_10,16,11,15 X_16,22,17,21 X₂₅₃₆ X_4,13,1,14
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -5, 3, -8, 7, -6, 11, -2, 8, -9, 5, -4, 6, -7, 9, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- u 5 -3 v u 4 + 7 u 4 + 10 v u 3 -14 u 3 -14 v u 2 + 10 u 2 + 7 v u -3 u - v$ (db)
Jones polynomial	$q^{5/2}-4 q^{3/2}+9 \sqrt{q}-\frac{16}{\sqrt{q}}+\frac{20}{q^{3/2}}-\frac{23}{q^{5/2}}+\frac{22}{q^{7/2}}-\frac{19}{q^{9/2}}+\frac{14}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 9 z -1 -4 z a 7 -3 a 7 z -1 + 6 z 3 a 5 + 9 z a 5 + 4 a 5 z -1 -3 z 5 a 3 -7 z 3 a 3 -8 z a 3 -2 a 3 z -1 - z 5 a + z 3 a + z a + z 3 a -1$ (db)
Kauffman polynomial	$-2 a 6 z 10 -2 a 4 z 10 -4 a 7 z 9 -13 a 5 z 9 -9 a 3 z 9 -3 a 8 z 8 -9 a 6 z 8 -22 a 4 z 8 -16 a 2 z 8 - a 9 z 7 + 7 a 7 z 7 + 22 a 5 z 7 - a 3 z 7 -15 a z 7 + 10 a 8 z 6 + 42 a 6 z 6 + 68 a 4 z 6 + 27 a 2 z 6 -9 z 6 + 4 a 9 z 5 + 8 a 7 z 5 + 19 a 5 z 5 + 41 a 3 z 5 + 22 a z 5 -4 z 5 a -1 -12 a 8 z 4 -44 a 6 z 4 -53 a 4 z 4 -14 a 2 z 4 - z 4 a -2 + 6 z 4 -6 a 9 z 3 -23 a 7 z 3 -44 a 5 z 3 -42 a 3 z 3 -14 a z 3 + z 3 a -1 + 6 a 8 z 2 + 17 a 6 z 2 + 15 a 4 z 2 + 3 a 2 z 2 - z 2 + 4 a 9 z + 15 a 7 z + 24 a 5 z + 16 a 3 z + 3 a z - a 8 -3 a 6 -2 a 4 - a 2 - a 9 z -1 -3 a 7 z -1 -4 a 5 z -1 -2 a 3 z -1$ (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 L11a127. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -8$	${\mathbb Z}$
$r = -7$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -5$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -4$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{9}$
$r = -3$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{11}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{10}$
$r = 1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 3$	${\mathbb Z}_2$	${\mathbb Z}$