Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)


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(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n60's page at Knotilus.

Visit L10n60's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n60's Link Presentations]

Planar diagram presentation X12,1,13,2 X16,7,17,8 X5,1,6,10 X3746 X9,5,10,4 X17,11,18,20 X13,19,14,18 X19,15,20,14 X2,11,3,12 X8,15,9,16
Gauss code {1, -9, -4, 5, -3, 4, 2, -10, -5, 3}, {9, -1, -7, 8, 10, -2, -6, 7, -8, 6}
A Braid Representative
A Morse Link Presentation L10n60 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1)^3-2 t(2)^2 t(1)^2+2 t(2) t(1)^2+2 t(2)^2 t(1)-2 t(2) t(1)-t(2)^3}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -q^{13/2}+2 q^{11/2}-2 q^{9/2}+3 q^{7/2}-2 q^{5/2}+2 q^{3/2}-2 \sqrt{q}-\frac{1}{\sqrt{q}}-\frac{1}{q^{5/2}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +9 z a^{-3} +5 a^{-3} z^{-1} -z^5 a^{-1} +a z^3-7 z^3 a^{-1} +4 a z-13 z a^{-1} +4 a z^{-1} -8 a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} -3 z^3 a^{-7} +z a^{-7} +2 z^6 a^{-6} -7 z^4 a^{-6} +5 z^2 a^{-6} -2 a^{-6} +z^7 a^{-5} -2 z^5 a^{-5} -2 z^3 a^{-5} +z a^{-5} + a^{-5} z^{-1} +3 z^6 a^{-4} -13 z^4 a^{-4} +18 z^2 a^{-4} -9 a^{-4} +z^7 a^{-3} -5 z^5 a^{-3} +9 z^3 a^{-3} -8 z a^{-3} +5 a^{-3} z^{-1} +2 z^6 a^{-2} -14 z^4 a^{-2} +28 z^2 a^{-2} -14 a^{-2} +a z^7+z^7 a^{-1} -7 a z^5-9 z^5 a^{-1} +15 a z^3+23 z^3 a^{-1} -13 a z-21 z a^{-1} +4 a z^{-1} +8 a^{-1} z^{-1} +z^6-8 z^4+15 z^2-8 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (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=1 is the signature of L10n60. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n60/KhovanovTable
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2 i=4
r=-4 {\mathbb Z} {\mathbb Z}
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{3} {\mathbb Z}^{2}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2} {\mathbb Z}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=6 {\mathbb Z}_2 {\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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