L11a41

(Knotscape image)

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Planar diagram presentation	X₆₁₇₂ X_10,4,11,3 X_22,15,5,16 X_16,7,17,8 X_20,18,21,17 X_14,10,15,9 X_12,19,13,20 X_18,13,19,14 X_8,21,9,22 X₂₅₃₆ X_4,12,1,11
Gauss code	{1, -10, 2, -11}, {10, -1, 4, -9, 6, -2, 11, -7, 8, -6, 3, -4, 5, -8, 7, -5, 9, -3}

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$\frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db)
Jones polynomial	$-\frac{15}{q^{9/2}}-q^{7/2}+\frac{21}{q^{7/2}}+5 q^{5/2}-\frac{26}{q^{5/2}}-12 q^{3/2}+\frac{26}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{8}{q^{11/2}}+18 \sqrt{q}-\frac{24}{\sqrt{q}}$ (db)
Signature	-1 (db)
HOMFLY-PT polynomial	$a 7 (- z)- a 7 z -1 + 3 a 5 z 3 + 5 a 5 z + 3 a 5 z -1 -3 a 3 z 5 -7 a 3 z 3 -7 a 3 z -3 a 3 z -1 + a z 7 + 3 a z 5 - z 5 a -1 + 5 a z 3 - z 3 a -1 + 4 a z + 2 a z -1 - z a -1 - a -1 z -1$ (db)
Kauffman polynomial	$a 8 z 6 -3 a 8 z 4 + 3 a 8 z 2 - a 8 + 3 a 7 z 7 -7 a 7 z 5 + 6 a 7 z 3 -3 a 7 z + a 7 z -1 + 5 a 6 z 8 -8 a 6 z 6 + 2 a 6 z 4 + 3 a 6 z 2 -2 a 6 + 5 a 5 z 9 - a 5 z 7 -15 a 5 z 5 + 21 a 5 z 3 -12 a 5 z + 3 a 5 z -1 + 2 a 4 z 10 + 15 a 4 z 8 -40 a 4 z 6 + 32 a 4 z 4 -8 a 4 z 2 + 14 a 3 z 9 -12 a 3 z 7 -23 a 3 z 5 + z 5 a -3 + 33 a 3 z 3 -16 a 3 z + 3 a 3 z -1 + 2 a 2 z 10 + 25 a 2 z 8 -58 a 2 z 6 + 5 z 6 a -2 + 39 a 2 z 4 -3 z 4 a -2 -11 a 2 z 2 + 2 a 2 + 9 a z 9 + 4 a z 7 + 12 z 7 a -1 -32 a z 5 -16 z 5 a -1 + 24 a z 3 + 6 z 3 a -1 -9 a z -2 z a -1 + 2 a z -1 + a -1 z -1 + 15 z 8 -22 z 6 + 9 z 4 -3 z 2$ (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 L11a41. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

$\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$	$i = -2$	$i = 0$
$r = -7$	${\mathbb Z}$
$r = -6$	${\mathbb Z}^{2}\oplus{\mathbb Z}_2$	${\mathbb Z}$
$r = -5$	${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2}$	${\mathbb Z}^{2}$
$r = -4$	${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6}$	${\mathbb Z}^{6}$
$r = -3$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9}$	${\mathbb Z}^{9}$
$r = -2$	${\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{12}$
$r = -1$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{14}$	${\mathbb Z}^{14}$
$r = 0$	${\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12}$	${\mathbb Z}^{14}$
$r = 1$	${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10}$	${\mathbb Z}^{10}$
$r = 2$	${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8}$	${\mathbb Z}^{8}$
$r = 3$	${\mathbb Z}\oplus{\mathbb Z}_2^{4}$	${\mathbb Z}^{4}$
$r = 4$	${\mathbb Z}_2$	${\mathbb Z}$