L11a135

(Knotscape image)

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

Planar diagram presentation	X₆₁₇₂ X_2,9,3,10 X_14,3,15,4 X_10,5,11,6 X_20,11,21,12 X_22,13,5,14 X_12,21,13,22 X_4,19,1,20 X_18,16,19,15 X_16,8,17,7 X_8,18,9,17
Gauss code	{1, -2, 3, -8}, {4, -1, 10, -11, 2, -4, 5, -7, 6, -3, 9, -10, 11, -9, 8, -5, 7, -6}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-\frac{2 t(1) t(2)^5-2 t(2)^5-6 t(1) t(2)^4+6 t(2)^4+10 t(1) t(2)^3-9 t(2)^3-9 t(1) t(2)^2+10 t(2)^2+6 t(1) t(2)-6 t(2)-2 t(1)+2}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$q^{3/2}-4 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{14}{q^{3/2}}+\frac{19}{q^{5/2}}-\frac{23}{q^{7/2}}+\frac{22}{q^{9/2}}-\frac{20}{q^{11/2}}+\frac{15}{q^{13/2}}-\frac{9}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$a 7 z 5 + 2 a 7 z 3 + a 7 z - a 5 z 7 -3 a 5 z 5 -3 a 5 z 3 - a 5 z + a 5 z -1 - a 3 z 7 -3 a 3 z 5 -3 a 3 z 3 -2 a 3 z - a 3 z -1 + a z 5 + 2 a z 3$ (db)
Kauffman polynomial	$- z 5 a 11 + z 3 a 11 -4 z 6 a 10 + 5 z 4 a 10 -2 z 2 a 10 -8 z 7 a 9 + 12 z 5 a 9 -7 z 3 a 9 + z a 9 -9 z 8 a 8 + 10 z 6 a 8 - z 4 a 8 - z 2 a 8 -6 z 9 a 7 -2 z 7 a 7 + 14 z 5 a 7 -6 z 3 a 7 -2 z 10 a 6 -13 z 8 a 6 + 27 z 6 a 6 -14 z 4 a 6 + 3 z 2 a 6 -12 z 9 a 5 + 19 z 7 a 5 -9 z 5 a 5 + 4 z 3 a 5 + z a 5 - a 5 z -1 -2 z 10 a 4 -11 z 8 a 4 + 32 z 6 a 4 -24 z 4 a 4 + 4 z 2 a 4 + a 4 -6 z 9 a 3 + 9 z 7 a 3 -4 z 3 a 3 + 2 z a 3 - a 3 z -1 -7 z 8 a 2 + 18 z 6 a 2 -14 z 4 a 2 + 2 z 2 a 2 -4 z 7 a + 10 z 5 a -6 z 3 a - z 6 + 2 z 4$ (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 L11a135. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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