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 L10n87's page at Knotilus.

Visit L10n87's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n87's 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}
A Braid Representative
A Morse Link Presentation L10n87 ML.gif

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 q^7-q^6+2 q^5-2 q^4+2 q^3-q^2+q+1+ q^{-2} (db)
Signature 2 (db)
HOMFLY-PT polynomial -z^6 a^{-2} -7 z^4 a^{-2} +z^4-14 z^2 a^{-2} +2 z^2 a^{-4} +z^2 a^{-6} +5 z^2-10 a^{-2} +3 a^{-4} + a^{-6} +6-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} (db)
Kauffman polynomial z^4 a^{-8} -3 z^2 a^{-8} + a^{-8} +z^5 a^{-7} -2 z^3 a^{-7} +z^6 a^{-6} -3 z^4 a^{-6} +3 z^2 a^{-6} - a^{-6} +z^5 a^{-5} -z^3 a^{-5} +z^4 a^{-4} -2 z^2 a^{-4} - a^{-4} z^{-2} +4 a^{-4} +z^7 a^{-3} -8 z^5 a^{-3} +19 z^3 a^{-3} -13 z a^{-3} +2 a^{-3} z^{-1} +z^8 a^{-2} -9 z^6 a^{-2} +26 z^4 a^{-2} -30 z^2 a^{-2} -2 a^{-2} z^{-2} +14 a^{-2} +z^7 a^{-1} -8 z^5 a^{-1} +18 z^3 a^{-1} -13 z a^{-1} +2 a^{-1} z^{-1} +z^8-8 z^6+21 z^4-22 z^2- z^{-2} +9 (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=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

(db, data source)

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

Back to the top.