L11a237

(Knotscape image)

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

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$- v 2 u 4 + 2 v u 4 - u 4 + 5 v 2 u 3 -9 v u 3 + 5 u 3 -8 v 2 u 2 + 15 v u 2 -8 u 2 + 5 v 2 u -9 v u + 5 u - v 2 + 2 v -1$ (db)
Jones polynomial	$-q^{13/2}+5 q^{11/2}-11 q^{9/2}+17 q^{7/2}-23 q^{5/2}+25 q^{3/2}-25 \sqrt{q}+\frac{20}{\sqrt{q}}-\frac{15}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{9/2}}$ (db)
Signature	1 (db)
HOMFLY-PT polynomial	$- z 7 a -1 + 2 a z 5 -3 z 5 a -1 + 2 z 5 a -3 - a 3 z 3 + 5 a z 3 -5 z 3 a -1 + 3 z 3 a -3 - z 3 a -5 -2 a 3 z + 6 a z -4 z a -1 + z a -3 - a 3 z -1 + 3 a z -1 -2 a -1 z -1$ (db)
Kauffman polynomial	$-2 z 10 a -2 -2 z 10 -5 a z 9 -13 z 9 a -1 -8 z 9 a -3 -5 a 2 z 8 -21 z 8 a -2 -13 z 8 a -4 -13 z 8 -3 a 3 z 7 + 2 a z 7 + 12 z 7 a -1 -4 z 7 a -3 -11 z 7 a -5 - a 4 z 6 + 8 a 2 z 6 + 46 z 6 a -2 + 17 z 6 a -4 -5 z 6 a -6 + 33 z 6 + 7 a 3 z 5 + 12 a z 5 + 16 z 5 a -1 + 27 z 5 a -3 + 15 z 5 a -5 - z 5 a -7 + 3 a 4 z 4 - a 2 z 4 -25 z 4 a -2 -4 z 4 a -4 + 4 z 4 a -6 -21 z 4 -6 a 3 z 3 -16 a z 3 -21 z 3 a -1 -16 z 3 a -3 -5 z 3 a -5 -3 a 4 z 2 -5 a 2 z 2 + 2 z 2 a -2 + 3 a 3 z + 10 a z + 9 z a -1 + 2 z a -3 + a 4 + 3 a 2 + 3- a 3 z -1 -3 a z -1 -2 a -1 z -1$ (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 L11a237. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 0$	$i = 2$
$r = -5$	${\mathbb Z}$
$r = -4$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -3$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -2$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -1$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 0$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{11}$	${\mathbb Z}^{12}$
$r = 1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 2$	${\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{13}$
$r = 3$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 4$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 5$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 6$	${\mathbb Z}_2$	${\mathbb Z}$