# L10n87

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## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n87's page at Knotilus. Visit L10n87's page at the original Knot Atlas.

### Link Presentations

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

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $-\frac{(w-1) \left(u v w^2+u w+u+v^2 w^3+v^2 w^2+v w\right)}{\sqrt{u} v w^2}$ (db) Jones polynomial q7−q6 + 2q5−2q4 + 2q3−q2 + q + 1 + q−2 (db) Signature 2 (db) HOMFLY-PT polynomial −z6a−2−7z4a−2 + z4−14z2a−2 + 2z2a−4 + z2a−6 + 5z2−10a−2 + 3a−4 + a−6 + 6−2a−2z−2 + a−4z−2 + z−2 (db) Kauffman polynomial z4a−8−3z2a−8 + a−8 + z5a−7−2z3a−7 + z6a−6−3z4a−6 + 3z2a−6−a−6 + z5a−5−z3a−5 + z4a−4−2z2a−4−a−4z−2 + 4a−4 + z7a−3−8z5a−3 + 19z3a−3−13za−3 + 2a−3z−1 + z8a−2−9z6a−2 + 26z4a−2−30z2a−2−2a−2z−2 + 14a−2 + z7a−1−8z5a−1 + 18z3a−1−13za−1 + 2a−1z−1 + z8−8z6 + 21z4−22z2−z−2 + 9 (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 = 2 is the signature of L10n87. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L10n87/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = 1 i = 3 i = 5 r = −4 ${\mathbb Z}$ ${\mathbb Z}$ r = −3 r = −2 ${\mathbb Z}$ r = −1 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 0 ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$ r = 1 ${\mathbb Z}$ ${\mathbb Z}$ ${\mathbb Z}$ r = 2 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = 3 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 4 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = 5 ${\mathbb Z}_2$ ${\mathbb Z}$ r = 6 ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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

###  Modifying This Page

Read me first: Modifying Knot Pages

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

See/edit the Link_Splice_Base (expert).

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