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 L10n55's page at Knotilus.

Visit L10n55's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n55's Link Presentations]

Planar diagram presentation X10,1,11,2 X11,17,12,16 X8,9,1,10 X17,9,18,20 X3,12,4,13 X7,14,8,15 X13,6,14,7 X5,18,6,19 X19,4,20,5 X15,2,16,3
Gauss code {1, 10, -5, 9, -8, 7, -6, -3}, {3, -1, -2, 5, -7, 6, -10, 2, -4, 8, -9, 4}
A Braid Representative
A Morse Link Presentation L10n55 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^3 (-v)+u^2 v^3-3 u^2 v^2+3 u^2 v-2 u^2-2 u v^3+3 u v^2-3 u v+u-v^2}{u^{3/2} v^{3/2}} (db)
Jones polynomial \frac{7}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{4}{q^{5/2}}-\frac{2}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{7}{q^{11/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z+a^9 z^{-1} -2 a^7 z^3-3 a^7 z-a^7 z^{-1} +a^5 z^5+2 a^5 z^3+a^5 z-2 a^3 z^3-3 a^3 z (db)
Kauffman polynomial a^{11} z^5-3 a^{11} z^3+2 a^{11} z+2 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+3 a^9 z^7-8 a^9 z^5+9 a^9 z^3-7 a^9 z+a^9 z^{-1} +a^8 z^8+3 a^8 z^6-10 a^8 z^4+6 a^8 z^2-a^8+5 a^7 z^7-11 a^7 z^5+12 a^7 z^3-7 a^7 z+a^7 z^{-1} +a^6 z^8+2 a^6 z^6-4 a^6 z^4+3 a^6 z^2+2 a^5 z^7-2 a^5 z^5+3 a^5 z^3-a^5 z+a^4 z^6+2 a^4 z^4-2 a^4 z^2+3 a^3 z^3-3 a^3 z (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 L10n55. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n55/KhovanovTable
Integral Khovanov Homology

(db, data source)

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