L10n68

Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n68's page at Knotilus. Visit L10n68's page at the original Knot Atlas.

 Planar diagram presentation X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X4,15,1,16 X19,10,20,5 Gauss code {1, 4, -3, -9}, {-2, -1, 5, 3, -6, 10}, {-7, 2, -4, -5, 9, 7, -8, 6, -10, 8}

Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{2 t(1) t(3)^2 t(2)^2-t(1) t(3) t(2)^2-t(1) t(3)^2 t(2)+t(1) t(3) t(2)-t(3) t(2)+t(2)+t(3)-2}{\sqrt{t(1)} t(2) t(3)}$ (db) Jones polynomial q−3−q−4 + 3q−5−2q−6 + 4q−7−3q−8 + 3q−9−2q−10 + q−11 (db) Signature -6 (db) HOMFLY-PT polynomial a12−a10z4−4a10z2 + a10z−2−2a10 + a8z6 + 4a8z4 + 2a8z2−2a8z−2−3a8 + a6z6 + 5a6z4 + 7a6z2 + a6z−2 + 4a6 (db) Kauffman polynomial a14z2−a14 + 2a13z3−2a13z + a12z6−3a12z4 + 5a12z2−a12 + 2a11z7−9a11z5 + 14a11z3−6a11z + a10z8−3a10z6 + a10z4 + 3a10z2 + a10z−2−3a10 + 3a9z7−12a9z5 + 11a9z3−2a9z−1 + a8z8−3a8z6−a8z4 + 6a8z2 + 2a8z−2−6a8 + a7z7−3a7z5−a7z3 + 4a7z−2a7z−1 + a6z6−5a6z4 + 7a6z2 + a6z−2−4a6 (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 = -6 is the signature of L10n68. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L10n68/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −7 i = −5 i = −3 r = −8 ${\mathbb Z}$ r = −7 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −6 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ r = −5 ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ ${\mathbb Z}$ r = −3 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ r = −1 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}$ ${\mathbb Z}$

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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