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 L10n89's page at Knotilus.

Visit L10n89's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n89's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X7,14,8,15 X20,15,17,16 X18,11,19,12 X12,17,13,18 X16,19,5,20 X13,8,14,9 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {6, -5, 7, -4}, {9, -1, -3, 8, 10, -2, 5, -6, -8, 3, 4, -7}
A Braid Representative
A Morse Link Presentation L10n89 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(3)-1) \left(t(1) t(3)^2+t(2) t(3)^2-t(3)^2-t(1) t(3)+2 t(1) t(2) t(3)-t(2) t(3)+2 t(3)+t(1)-t(1) t(2)+t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial 2 q^{-10} -4 q^{-9} +7 q^{-8} -8 q^{-7} +9 q^{-6} -7 q^{-5} +7 q^{-4} -3 q^{-3} + q^{-2} (db)
Signature -4 (db)
HOMFLY-PT polynomial a^{10} z^{-2} +2 a^{10}-5 z^2 a^8-2 a^8 z^{-2} -8 a^8+3 z^4 a^6+8 z^2 a^6+a^6 z^{-2} +6 a^6+z^4 a^4+z^2 a^4 (db)
Kauffman polynomial 3 a^{12} z^4-5 a^{12} z^2+2 a^{12}+a^{11} z^7+a^{11} z^5-2 a^{11} z^3+a^{10} z^8+2 a^{10} z^6-3 a^{10} z^4+a^{10} z^{-2} -2 a^{10}+5 a^9 z^7-3 a^9 z^5-8 a^9 z^3+8 a^9 z-2 a^9 z^{-1} +a^8 z^8+8 a^8 z^6-18 a^8 z^4+16 a^8 z^2+2 a^8 z^{-2} -9 a^8+4 a^7 z^7-a^7 z^5-8 a^7 z^3+8 a^7 z-2 a^7 z^{-1} +6 a^6 z^6-11 a^6 z^4+10 a^6 z^2+a^6 z^{-2} -6 a^6+3 a^5 z^5-2 a^5 z^3+a^4 z^4-a^4 z^2 (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=-4 is the signature of L10n89. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n89/KhovanovTable
Integral Khovanov Homology

(db, data source)

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