# L10n74

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## Contents

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n74's page at Knotilus. Visit L10n74's page at the original Knot Atlas.

### Link Presentations

 Planar diagram presentation X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,12,20,5 X11,20,12,13 X13,10,14,11 Gauss code {1, 4, -3, -7}, {-2, -1, 5, 3, -6, 10, -9, 8}, {-10, 2, -4, -5, 7, 6, -8, 9}

### Polynomial invariants

 Multivariable Alexander Polynomial (in u, v, w, ...) $\frac{u v^3 w^2+u v^2 w^3-u v^2 w^2+v w-v-w}{\sqrt{u} v^{3/2} w^{3/2}}$ (db) Jones polynomial q−3−q−4 + 2q−5−q−6 + 2q−7−2q−8 + 2q−9 + q−11 (db) Signature -6 (db) HOMFLY-PT polynomial a12z−2 + a12−z4a10−5z2a10−2a10z−2−5a10 + z6a8 + 5z4a8 + 6z2a8 + a8z−2 + 2a8 + z6a6 + 5z4a6 + 6z2a6 + 2a6 (db) Kauffman polynomial a14z2−2a14 + a13z + a12z4−3a12z2−a12z−2 + 3a12 + a11z7−6a11z5 + 12a11z3−8a11z + 2a11z−1 + a10z8−6a10z6 + 13a10z4−15a10z2−2a10z−2 + 9a10 + 2a9z7−10a9z5 + 14a9z3−8a9z + 2a9z−1 + a8z8−5a8z6 + 7a8z4−5a8z2−a8z−2 + 3a8 + a7z7−4a7z5 + 2a7z3 + a7z + a6z6−5a6z4 + 6a6z2−2a6 (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 L10n74. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L10n74/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ i = −7 i = −5 i = −3 r = −8 ${\mathbb Z}$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ r = −7 ${\mathbb Z}$ r = −6 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ r = −5 ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −4 ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ r = −3 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ r = −2 ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ 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.

###  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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