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 L10n67's page at Knotilus.

Visit L10n67's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n67's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X11,16,12,17 X13,19,14,18 X17,20,18,9 X19,13,20,12 X8,16,5,15 X14,8,15,7 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {9, -1, 8, -7}, {10, -2, -3, 6, -4, -8, 7, 3, -5, 4, -6, 5}
A Braid Representative
A Morse Link Presentation L10n67 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2) t(3)^3-2 t(1) t(3)^2+t(1) t(2) t(3)^2-4 t(2) t(3)^2+t(3)^2+4 t(1) t(3)-t(1) t(2) t(3)+2 t(2) t(3)-t(3)-t(1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial 2 q^2-3 q+6-6 q^{-1} +6 q^{-2} -5 q^{-3} +5 q^{-4} -2 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 z^{-2} +a^6-3 z^2 a^4-3 a^4 z^{-2} -5 a^4+2 z^4 a^2+6 z^2 a^2+4 a^2 z^{-2} +8 a^2-4 z^2-3 z^{-2} -6+ a^{-2} z^{-2} +2 a^{-2} (db)
Kauffman polynomial a^6 z^6-4 a^6 z^4+6 a^6 z^2+a^6 z^{-2} -4 a^6+2 a^5 z^7-6 a^5 z^5+3 a^5 z^3+a^5 z-a^5 z^{-1} +a^4 z^8+3 a^4 z^6-21 a^4 z^4+25 a^4 z^2+3 a^4 z^{-2} -14 a^4+6 a^3 z^7-18 a^3 z^5+11 a^3 z^3+a^3 z-a^3 z^{-1} +a^2 z^8+6 a^2 z^6-30 a^2 z^4+39 a^2 z^2+3 z^2 a^{-2} +4 a^2 z^{-2} + a^{-2} z^{-2} -21 a^2-4 a^{-2} +4 a z^7-11 a z^5+z^5 a^{-1} +9 a z^3+z^3 a^{-1} +a z+z a^{-1} -a z^{-1} - a^{-1} z^{-1} +4 z^6-13 z^4+23 z^2+3 z^{-2} -14 (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=0 is the signature of L10n67. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n67/KhovanovTable
Integral Khovanov Homology

(db, data source)

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

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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