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 L10n64's page at Knotilus.

Visit L10n64's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n64's Link Presentations]

Planar diagram presentation X12,1,13,2 X16,7,17,8 X3948 X17,2,18,3 X14,6,15,5 X6,12,7,11 X9,18,10,19 X20,15,11,16 X10,13,1,14 X4,19,5,20
Gauss code {1, 4, -3, -10, 5, -6, 2, 3, -7, -9}, {6, -1, 9, -5, 8, -2, -4, 7, 10, -8}
A Braid Representative
A Morse Link Presentation L10n64 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^3 \left(-v^2\right)+2 u^3 v-u^3+u^2 v^2-3 u^2 v+u^2+u v^3-3 u v^2+u v-v^3+2 v^2-v}{u^{3/2} v^{3/2}} (db)
Jones polynomial \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)+2 a^5 z^3+3 a^5 z+a^5 z^{-1} -a^3 z^5-3 a^3 z^3-4 a^3 z-a^3 z^{-1} +a z^3 (db)
Kauffman polynomial a^8 z^6-3 a^8 z^4+2 a^8 z^2+3 a^7 z^7-11 a^7 z^5+10 a^7 z^3-2 a^7 z+2 a^6 z^8-4 a^6 z^6-3 a^6 z^4+3 a^6 z^2+7 a^5 z^7-24 a^5 z^5+23 a^5 z^3-9 a^5 z+a^5 z^{-1} +2 a^4 z^8-3 a^4 z^6-2 a^4 z^4+2 a^4 z^2-a^4+4 a^3 z^7-13 a^3 z^5+17 a^3 z^3-8 a^3 z+a^3 z^{-1} +2 a^2 z^6-2 a^2 z^4+2 a^2 z^2+4 a z^3-a z+z^2 (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 L10n64. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n64/KhovanovTable
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\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).

Back to the top.