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 L10n91's page at Knotilus.

Visit L10n91's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n91's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X13,20,14,17 X7,18,8,19 X17,10,18,11 X9,15,10,14 X15,9,16,8 X19,16,20,5 X2536 X4,11,1,12
Gauss code {1, -9, 2, -10}, {-5, 4, -8, 3}, {9, -1, -4, 7, -6, 5, 10, -2, -3, 6, -7, 8}
A Braid Representative
A Morse Link Presentation L10n91 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(3)-1) (t(3)+1) (-2 t(1) t(2)+t(1) t(3) t(2)-2 t(3)+1)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial 1- q^{-1} + q^{-2} - q^{-3} + q^{-5} + q^{-6} +2 q^{-7} - q^{-8} + q^{-9} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^{10} z^{-2} -2 a^8 z^{-2} -4 a^8+z^4 a^6+6 z^2 a^6+a^6 z^{-2} +6 a^6-z^6 a^4-5 z^4 a^4-6 z^2 a^4-4 a^4+z^4 a^2+4 z^2 a^2+2 a^2 (db)
Kauffman polynomial a^{10} z^6-5 a^{10} z^4+6 a^{10} z^2+a^{10} z^{-2} -4 a^{10}+a^9 z^7-4 a^9 z^5+4 a^9 z-2 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+3 a^8 z^4+6 a^8 z^2+2 a^8 z^{-2} -5 a^8+2 a^7 z^7-12 a^7 z^5+16 a^7 z^3-4 a^7 z-2 a^7 z^{-1} +a^6 z^8-5 a^6 z^6+2 a^6 z^4+6 a^6 z^2+a^6 z^{-2} -2 a^6+2 a^5 z^7-13 a^5 z^5+22 a^5 z^3-12 a^5 z+2 a^4 z^6-11 a^4 z^4+12 a^4 z^2-2 a^4+a^3 z^7-5 a^3 z^5+6 a^3 z^3-4 a^3 z+a^2 z^6-5 a^2 z^4+6 a^2 z^2-2 a^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=-3 is the signature of L10n91. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n91/KhovanovTable
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3 i=-1
r=-8 {\mathbb Z} {\mathbb Z}
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3} {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}
r=2 {\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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