L10a17

(Knotscape image)

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

A Braid Representative

A Morse Link Presentation

Multivariable Alexander Polynomial (in $u$ , $v$ , $w$ , ...)	$-3 v u 3 + 3 u 3 + 7 v u 2 -7 u 2 -7 v u + 7 u + 3 v -3$ (db)
Jones polynomial	$-q^{19/2}+4 q^{17/2}-7 q^{15/2}+11 q^{13/2}-13 q^{11/2}+13 q^{9/2}-13 q^{7/2}+9 q^{5/2}-6 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db)
Signature	3 (db)
HOMFLY-PT polynomial	$- z 5 a -3 -2 z 5 a -5 + z 3 a -1 - z 3 a -3 -5 z 3 a -5 + 3 z 3 a -7 + 2 z a -1 -5 z a -5 + 4 z a -7 - z a -9 + a -1 z -1 -2 a -5 z -1 + a -7 z -1$ (db)
Kauffman polynomial	$- z 9 a -5 - z 9 a -7 -3 z 8 a -4 -7 z 8 a -6 -4 z 8 a -8 -3 z 7 a -3 -9 z 7 a -5 -12 z 7 a -7 -6 z 7 a -9 -2 z 6 a -2 + z 6 a -4 + 5 z 6 a -6 -2 z 6 a -8 -4 z 6 a -10 - z 5 a -1 + 3 z 5 a -3 + 22 z 5 a -5 + 29 z 5 a -7 + 10 z 5 a -9 - z 5 a -11 + 3 z 4 a -2 + 3 z 4 a -4 + 10 z 4 a -6 + 17 z 4 a -8 + 7 z 4 a -10 + 3 z 3 a -1 -21 z 3 a -5 -23 z 3 a -7 -4 z 3 a -9 + z 3 a -11 -5 z 2 a -4 -14 z 2 a -6 -12 z 2 a -8 -3 z 2 a -10 -3 z a -1 + 9 z a -5 + 8 z a -7 + 2 z a -9 - a -2 + 3 a -4 + 5 a -6 + 2 a -8 + a -1 z -1 -2 a -5 z -1 - a -7 z -1$ (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 L10a17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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