L9n5

Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n5's page at Knotilus. Visit L9n5's page at the original Knot Atlas. L9n5 is $9^2_{44}$ in the Rolfsen table of links.

 Planar diagram presentation X6172 X14,7,15,8 X15,1,16,4 X5,12,6,13 X3849 X9,16,10,17 X11,18,12,5 X17,10,18,11 X2,14,3,13 Gauss code {1, -9, -5, 3}, {-4, -1, 2, 5, -6, 8, -7, 4, 9, -2, -3, 6, -8, 7}

Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $\frac{3}{q^{9/2}}-\frac{3}{q^{7/2}}+\frac{1}{q^{5/2}}-\frac{2}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial −z3a7−3za7−2a7z−1 + z5a5 + 5z3a5 + 9za5 + 5a5z−1−2z3a3−6za3−3a3z−1 (db) Kauffman polynomial a10z4−3a10z2 + a10 + a9z5−2a9z3 + a8z6−2a8z4 + a8z2 + a7z7−4a7z5 + 8a7z3−5a7z + 2a7z−1 + 2a6z6−7a6z4 + 12a6z2−5a6 + a5z7−5a5z5 + 13a5z3−13a5z + 5a5z−1 + a4z6−4a4z4 + 8a4z2−5a4 + 3a3z3−8a3z + 3a3z−1 (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 = -3 is the signature of L9n5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L9n5/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −4 i = −2 r = −7 ${\mathbb Z}$ r = −6 ${\mathbb Z}_2$ ${\mathbb Z}$ r = −5 ${\mathbb Z}^{2}$ r = −4 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ 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 ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$

Computer Talk

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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).