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)


From Knot Atlas
Jump to: navigation, search






(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n68's page at Knotilus.

Visit L10n68's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n68's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X4,15,1,16 X19,10,20,5
Gauss code {1, 4, -3, -9}, {-2, -1, 5, 3, -6, 10}, {-7, 2, -4, -5, 9, 7, -8, 6, -10, 8}
A Braid Representative
A Morse Link Presentation L10n68 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(1) t(3)^2 t(2)^2-t(1) t(3) t(2)^2-t(1) t(3)^2 t(2)+t(1) t(3) t(2)-t(3) t(2)+t(2)+t(3)-2}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial  q^{-3} - q^{-4} +3 q^{-5} -2 q^{-6} +4 q^{-7} -3 q^{-8} +3 q^{-9} -2 q^{-10} + q^{-11} (db)
Signature -6 (db)
HOMFLY-PT polynomial a^{12}-a^{10} z^4-4 a^{10} z^2+a^{10} z^{-2} -2 a^{10}+a^8 z^6+4 a^8 z^4+2 a^8 z^2-2 a^8 z^{-2} -3 a^8+a^6 z^6+5 a^6 z^4+7 a^6 z^2+a^6 z^{-2} +4 a^6 (db)
Kauffman polynomial a^{14} z^2-a^{14}+2 a^{13} z^3-2 a^{13} z+a^{12} z^6-3 a^{12} z^4+5 a^{12} z^2-a^{12}+2 a^{11} z^7-9 a^{11} z^5+14 a^{11} z^3-6 a^{11} z+a^{10} z^8-3 a^{10} z^6+a^{10} z^4+3 a^{10} z^2+a^{10} z^{-2} -3 a^{10}+3 a^9 z^7-12 a^9 z^5+11 a^9 z^3-2 a^9 z^{-1} +a^8 z^8-3 a^8 z^6-a^8 z^4+6 a^8 z^2+2 a^8 z^{-2} -6 a^8+a^7 z^7-3 a^7 z^5-a^7 z^3+4 a^7 z-2 a^7 z^{-1} +a^6 z^6-5 a^6 z^4+7 a^6 z^2+a^6 z^{-2} -4 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 L10n68. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n68/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}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{3} {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4} {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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.