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 L10n90's page at Knotilus.

Visit L10n90's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n90's Link Presentations]

Planar diagram presentation X6172 X3,13,4,12 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 X11,1,12,4
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 L10n90 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(3)-1) \left(t(2) t(3)^2-2 t(2) t(3)+2 t(3)-1\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial q^3-3 q^2+5 q-7+9 q^{-1} -7 q^{-2} +8 q^{-3} -5 q^{-4} +3 q^{-5} (db)
Signature -2 (db)
HOMFLY-PT polynomial a^6 z^{-2} +a^6-z^4 a^4-3 z^2 a^4-2 a^4 z^{-2} -5 a^4+z^6 a^2+4 z^4 a^2+7 z^2 a^2+a^2 z^{-2} +6 a^2-2 z^4-5 z^2-3+z^2 a^{-2} + a^{-2} (db)
Kauffman polynomial 2 a^2 z^8+2 z^8+6 a^3 z^7+9 a z^7+3 z^7 a^{-1} +7 a^4 z^6+5 a^2 z^6+z^6 a^{-2} -z^6+3 a^5 z^5-12 a^3 z^5-25 a z^5-10 z^5 a^{-1} -15 a^4 z^4-23 a^2 z^4-3 z^4 a^{-2} -11 z^4+7 a^3 z^3+16 a z^3+9 z^3 a^{-1} +6 a^6 z^2+16 a^4 z^2+18 a^2 z^2+3 z^2 a^{-2} +11 z^2+3 a^5 z+a^3 z-3 a z-z a^{-1} -4 a^6-8 a^4-7 a^2- a^{-2} -3-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^2 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=-2 is the signature of L10n90. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n90/KhovanovTable
Integral Khovanov Homology

(db, data source)

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