L11a234

(Knotscape image)

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

Planar diagram presentation	X₈₁₉₂ X_12,4,13,3 X_18,6,19,5 X_22,12,7,11 X_20,17,21,18 X_16,21,17,22 X_14,10,15,9 X_10,16,11,15 X_4,20,5,19 X₂₇₃₈ X_6,14,1,13
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 7, -8, 4, -2, 11, -7, 8, -6, 5, -3, 9, -5, 6, -4}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 3 v u 4 -2 u 4 + 3 v 2 u 3 -8 v u 3 + 4 u 3 -4 v 2 u 2 + 9 v u 2 -4 u 2 + 4 v 2 u -8 v u + 3 u -2 v 2 + 3 v -1$ (db)
Jones polynomial	$q^{17/2}-3 q^{15/2}+7 q^{13/2}-12 q^{11/2}+16 q^{9/2}-19 q^{7/2}+18 q^{5/2}-17 q^{3/2}+12 \sqrt{q}-\frac{8}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$z 7 a -3 -2 z 5 a -1 + 4 z 5 a -3 -2 z 5 a -5 + a z 3 -5 z 3 a -1 + 8 z 3 a -3 -6 z 3 a -5 + z 3 a -7 + a z -3 z a -1 + 8 z a -3 -7 z a -5 + 2 z a -7 + 2 a -3 z -1 -3 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 a -4 -5 z 9 a -1 -11 z 9 a -3 -6 z 9 a -5 -6 z 8 a -2 -11 z 8 a -4 -9 z 8 a -6 -4 z 8 - a z 7 + 14 z 7 a -1 + 25 z 7 a -3 + z 7 a -5 -9 z 7 a -7 + 34 z 6 a -2 + 38 z 6 a -4 + 12 z 6 a -6 -6 z 6 a -8 + 14 z 6 + 3 a z 5 -6 z 5 a -1 -3 z 5 a -3 + 23 z 5 a -5 + 14 z 5 a -7 -3 z 5 a -9 -32 z 4 a -2 -26 z 4 a -4 - z 4 a -6 + 6 z 4 a -8 - z 4 a -10 -14 z 4 -3 a z 3 -6 z 3 a -1 -15 z 3 a -3 -27 z 3 a -5 -13 z 3 a -7 + 2 z 3 a -9 + 7 z 2 a -2 + 2 z 2 a -4 -7 z 2 a -6 -4 z 2 a -8 + z 2 a -10 + 3 z 2 + a z + 3 z a -1 + 10 z a -3 + 13 z a -5 + 5 z a -7 + 3 a -4 + 3 a -6 + a -8 -2 a -3 z -1 -3 a -5 z -1 - a -7 z -1$ (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 =

3 is the signature of L11a234. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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