L11a233

(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_20,12,21,11 X_22,16,7,15 X_10,18,11,17 X_16,22,17,21 X_14,10,15,9 X_4,20,5,19 X₂₇₃₈ X_6,14,1,13
Gauss code	{1, -10, 2, -9, 3, -11}, {10, -1, 8, -6, 4, -2, 11, -8, 5, -7, 6, -3, 9, -4, 7, -5}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{3 u^2 v^3-7 u^2 v^2+6 u^2 v-2 u^2+2 u v^4-11 u v^3+17 u v^2-11 u v+2 u-2 v^4+6 v^3-7 v^2+3 v}{u v^2}$ (db)
Jones polynomial	$q^{21/2}-4 q^{19/2}+10 q^{17/2}-16 q^{15/2}+22 q^{13/2}-26 q^{11/2}+25 q^{9/2}-23 q^{7/2}+16 q^{5/2}-10 q^{3/2}+4 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 5 a -3 -3 z 5 a -5 - z 5 a -7 + z 3 a -1 + 2 z 3 a -3 -5 z 3 a -5 + z 3 a -7 + z 3 a -9 + 4 z a -3 -3 z a -5 + a -3 z -1 -2 a -7 z -1 + a -9 z -1$ (db)
Kauffman polynomial	$-3 z 10 a -6 -3 z 10 a -8 -9 z 9 a -5 -17 z 9 a -7 -8 z 9 a -9 -12 z 8 a -4 -17 z 8 a -6 -13 z 8 a -8 -8 z 8 a -10 -9 z 7 a -3 + 5 z 7 a -5 + 31 z 7 a -7 + 13 z 7 a -9 -4 z 7 a -11 -4 z 6 a -2 + 20 z 6 a -4 + 48 z 6 a -6 + 43 z 6 a -8 + 18 z 6 a -10 - z 6 a -12 - z 5 a -1 + 14 z 5 a -3 + 14 z 5 a -5 -14 z 5 a -7 -5 z 5 a -9 + 8 z 5 a -11 + 4 z 4 a -2 -14 z 4 a -4 -41 z 4 a -6 -39 z 4 a -8 -14 z 4 a -10 + 2 z 4 a -12 + z 3 a -1 -10 z 3 a -3 -18 z 3 a -5 + z 3 a -7 + 4 z 3 a -9 -4 z 3 a -11 + 3 z 2 a -4 + 14 z 2 a -6 + 19 z 2 a -8 + 7 z 2 a -10 - z 2 a -12 + 5 z a -3 + 5 z a -5 -2 z a -7 -2 z a -9 + a -4 -3 a -6 -5 a -8 -2 a -10 - a -3 z -1 + 2 a -7 z -1 + a -9 z -1$ (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 L11a233. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = 2$	$i = 4$
$r = -2$	${\mathbb Z}$
$r = -1$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = 0$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{4}$
$r = 1$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = 2$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{11}$
$r = 3$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = 4$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 5$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = 6$	${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = 7$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7}$	${\mathbb Z}^{7}$
$r = 8$	${\mathbb Z}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = 9$	${\mathbb Z}_2$	${\mathbb Z}$