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 L10n93's page at Knotilus.

Visit L10n93's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n93's Link Presentations]

Planar diagram presentation X8192 X9,18,10,19 X17,6,18,1 X16,7,17,8 X3,10,4,11 X5,14,6,15 X13,4,14,5 X20,11,13,12 X12,15,7,16 X2,19,3,20
Gauss code {1, -10, -5, 7, -6, 3}, {4, -1, -2, 5, 8, -9}, {-7, 6, 9, -4, -3, 2, 10, -8}
A Braid Representative
A Morse Link Presentation L10n93 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u v w+1) \left(u v w^2-1\right)}{u v w^{3/2}} (db)
Jones polynomial  q^{-11} + q^{-9} + q^{-6} + q^{-4} (db)
Signature -7 (db)
HOMFLY-PT polynomial z^2 a^{12}+a^{12} z^{-2} +2 a^{12}-z^6 a^{10}-7 z^4 a^{10}-14 z^2 a^{10}-2 a^{10} z^{-2} -9 a^{10}+z^8 a^8+8 z^6 a^8+21 z^4 a^8+21 z^2 a^8+a^8 z^{-2} +7 a^8 (db)
Kauffman polynomial a^{14} z^2-2 a^{14}-a^{12} z^2-a^{12} z^{-2} +3 a^{12}+a^{11} z^7-7 a^{11} z^5+14 a^{11} z^3-9 a^{11} z+2 a^{11} z^{-1} +a^{10} z^8-8 a^{10} z^6+21 a^{10} z^4-23 a^{10} z^2-2 a^{10} z^{-2} +11 a^{10}+a^9 z^7-7 a^9 z^5+14 a^9 z^3-9 a^9 z+2 a^9 z^{-1} +a^8 z^8-8 a^8 z^6+21 a^8 z^4-21 a^8 z^2-a^8 z^{-2} +7 a^8 (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=-7 is the signature of L10n93. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n93/KhovanovTable
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-9 i=-7 i=-5
r=-8 {\mathbb Z} {\mathbb Z}
r=-6 {\mathbb Z}^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z} {\mathbb Z}
r=-4 {\mathbb Z} {\mathbb Z}
r=-3 {\mathbb Z}
r=-2 {\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.