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 L10n73's page at Knotilus.

Visit L10n73's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n73's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X7,17,8,16 X20,9,11,10 X18,12,19,11 X15,9,16,8 X10,19,5,20 X17,14,18,15 X2536 X4,14,1,13
Gauss code {1, -9, 2, -10}, {9, -1, -3, 6, 4, -7}, {5, -2, 10, 8, -6, 3, -8, -5, 7, -4}
A Braid Representative
A Morse Link Presentation L10n73 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v+w-1) (v w-v-w)}{\sqrt{u} v w} (db)
Jones polynomial -q^5+3 q^4-4 q^3- q^{-3} +6 q^2+4 q^{-2} -6 q-4 q^{-1} +7 (db)
Signature 0 (db)
HOMFLY-PT polynomial z^4 a^{-2} +z^4-a^2 z^2+z^2 a^{-2} -z^2 a^{-4} +a^2+ a^{-2} -2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial z^5 a^{-5} -2 z^3 a^{-5} +3 z^6 a^{-4} -8 z^4 a^{-4} +4 z^2 a^{-4} +3 z^7 a^{-3} -7 z^5 a^{-3} +a^3 z^3+3 z^3 a^{-3} +z^8 a^{-2} +2 z^6 a^{-2} +4 a^2 z^4-8 z^4 a^{-2} -3 a^2 z^2+5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} -2 a^2-2 a^{-2} +a z^7+4 z^7 a^{-1} -8 z^5 a^{-1} +a z^3+5 z^3 a^{-1} +2 a z+2 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +z^8-z^6+4 z^4-2 z^2+2 z^{-2} -3 (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=0 is the signature of L10n73. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n73/KhovanovTable
Integral Khovanov Homology

(db, data source)

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

Back to the top.