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 L10n74's page at Knotilus.

Visit L10n74's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n74's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X3849 X15,2,16,3 X16,7,17,8 X9,18,10,19 X4,17,1,18 X19,12,20,5 X11,20,12,13 X13,10,14,11
Gauss code {1, 4, -3, -7}, {-2, -1, 5, 3, -6, 10, -9, 8}, {-10, 2, -4, -5, 7, 6, -8, 9}
A Braid Representative
A Morse Link Presentation L10n74 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u v^3 w^2+u v^2 w^3-u v^2 w^2+v w-v-w}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial  q^{-3} - q^{-4} +2 q^{-5} - q^{-6} +2 q^{-7} -2 q^{-8} +2 q^{-9} + q^{-11} (db)
Signature -6 (db)
HOMFLY-PT polynomial a^{12} z^{-2} +a^{12}-z^4 a^{10}-5 z^2 a^{10}-2 a^{10} z^{-2} -5 a^{10}+z^6 a^8+5 z^4 a^8+6 z^2 a^8+a^8 z^{-2} +2 a^8+z^6 a^6+5 z^4 a^6+6 z^2 a^6+2 a^6 (db)
Kauffman polynomial a^{14} z^2-2 a^{14}+a^{13} z+a^{12} z^4-3 a^{12} z^2-a^{12} z^{-2} +3 a^{12}+a^{11} z^7-6 a^{11} z^5+12 a^{11} z^3-8 a^{11} z+2 a^{11} z^{-1} +a^{10} z^8-6 a^{10} z^6+13 a^{10} z^4-15 a^{10} z^2-2 a^{10} z^{-2} +9 a^{10}+2 a^9 z^7-10 a^9 z^5+14 a^9 z^3-8 a^9 z+2 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+7 a^8 z^4-5 a^8 z^2-a^8 z^{-2} +3 a^8+a^7 z^7-4 a^7 z^5+2 a^7 z^3+a^7 z+a^6 z^6-5 a^6 z^4+6 a^6 z^2-2 a^6 (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=-6 is the signature of L10n74. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n74/KhovanovTable
Integral Khovanov Homology

(db, data source)

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