L10n88

Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n88's page at Knotilus. Visit L10n88's page at the original Knot Atlas.

 Planar diagram presentation X6172 X10,3,11,4 X7,14,8,15 X15,17,16,20 X11,19,12,18 X17,13,18,12 X19,5,20,16 X13,8,14,9 X2536 X4,9,1,10 Gauss code {1, -9, 2, -10}, {-6, 5, -7, 4}, {9, -1, -3, 8, 10, -2, -5, 6, -8, 3, -4, 7}
A Braid Representative

Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{(t(3)-1) \left(-t(1) t(3)^2+t(2) t(3)^2-t(3)^2-t(1) t(3)-t(2) t(3)+t(1)-t(1) t(2)-t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial q3 + 3−2q−1 + 3q−2−3q−3 + 3q−4−2q−5 + q−6 (db) Signature 0 (db) HOMFLY-PT polynomial a6−2z2a4−2a4 + z4a2 + 3z2a2 + a2z−2 + 4a2−z4−6z2−2z−2−6 + z2a−2 + a−2z−2 + 3a−2 (db) Kauffman polynomial a4z8 + a2z8 + 2a5z7 + 3a3z7 + az7 + a6z6−2a4z6−4a2z6 + z6a−2−8a5z5−13a3z5−5az5−4a6z4−4a4z4 + 3a2z4−6z4a−2−3z4 + 7a5z3 + 15a3z3 + 5az3−3z3a−1 + 4a6z2 + 4a4z2 + a2z2 + 9z2a−2 + 10z2−2a5z−6a3z + 2az + 6za−1−a6−2a2−5a−2−7−2az−1−2a−1z−1 + a2z−2 + a−2z−2 + 2z−2 (db)

Khovanov Homology

 The coefficients of the monomials trqj 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−2r = s + 1 or j−2r = s−1, where s = 0 is the signature of L10n88. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
\ r
\
j \
-6-5-4-3-2-101234χ
7          11
5          11
3       11  0
1      3    3
-1     241   1
-3    21     1
-5   121     0
-7  22       0
-9 12        1
-11 1         -1
-131          1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −3 i = −1 i = 1 r = −6 ${\mathbb Z}$ r = −5 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = −3 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −1 ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = 0 ${\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ r = 1 ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ r = 2 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 3 r = 4 ${\mathbb Z}$ ${\mathbb Z}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.

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