L11a11

(Knotscape image)

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

Visit L11a11's page at the original Knot Atlas.

Planar diagram presentation	X₆₁₇₂ X_20,7,21,8 X_4,21,1,22 X_14,10,15,9 X₈₄₉₃ X_12,5,13,6 X_22,13,5,14 X_18,16,19,15 X_16,11,17,12 X_10,17,11,18 X_2,20,3,19
Gauss code	{1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 9, -6, 7, -4, 8, -9, 10, -8, 11, -2, 3, -7}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-5 t(2)^3+9 t(2)^2-5 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{16}{q^{9/2}}-q^{7/2}+\frac{23}{q^{7/2}}+5 q^{5/2}-\frac{27}{q^{5/2}}-12 q^{3/2}+\frac{27}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{9}{q^{11/2}}+18 \sqrt{q}-\frac{25}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 (- z) + 3 a 5 z 3 + 3 a 5 z -3 a 3 z 5 -6 a 3 z 3 -4 a 3 z + a z 7 + 3 a z 5 - z 5 a -1 + 5 a z 3 - z 3 a -1 + 3 a z + a z -1 - z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$a 8 z 6 -2 a 8 z 4 + a 8 z 2 + 4 a 7 z 7 -9 a 7 z 5 + 7 a 7 z 3 -2 a 7 z + 7 a 6 z 8 -14 a 6 z 6 + 10 a 6 z 4 -3 a 6 z 2 + 6 a 5 z 9 -22 a 5 z 5 + 23 a 5 z 3 -6 a 5 z + 2 a 4 z 10 + 20 a 4 z 8 -53 a 4 z 6 + 41 a 4 z 4 -10 a 4 z 2 + 15 a 3 z 9 -9 a 3 z 7 -34 a 3 z 5 + z 5 a -3 + 35 a 3 z 3 -8 a 3 z + 2 a 2 z 10 + 28 a 2 z 8 -64 a 2 z 6 + 5 z 6 a -2 + 41 a 2 z 4 -3 z 4 a -2 -9 a 2 z 2 + 9 a z 9 + 7 a z 7 + 12 z 7 a -1 -38 a z 5 -16 z 5 a -1 + 26 a z 3 + 7 z 3 a -1 -6 a z -2 z a -1 + a z -1 + a -1 z -1 + 15 z 8 -21 z 6 + 9 z 4 -3 z 2 -1$ (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 L11a11. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{3}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$	${\mathbb Z}^{3}$
$r = -4$	${\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = -2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{13}$
$r = -1$	${\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13}$	${\mathbb Z}^{15}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$