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 L10n95's page at Knotilus.

Visit L10n95's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n95's Link Presentations]

Planar diagram presentation X8192 X18,12,19,11 X3,10,4,11 X19,2,20,3 X16,8,17,7 X9,13,10,20 X12,18,7,17 X4,13,5,14 X14,5,15,6 X6,15,1,16
Gauss code {1, 4, -3, -8, 9, -10}, {5, -1, -6, 3, 2, -7}, {8, -9, 10, -5, 7, -2, -4, 6}
A Braid Representative
A Morse Link Presentation L10n95 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1) t(2) t(3)^3+t(1)^2 t(3)^2-t(1) t(2)^2 t(3)^2-t(1) t(3)^2-t(1)^2 t(2) t(3)^2+2 t(1) t(2) t(3)^2-t(2) t(3)^2+t(1) t(2)^2 t(3)-t(2)^2 t(3)+t(1) t(3)+t(1)^2 t(2) t(3)-2 t(1) t(2) t(3)+t(2) t(3)+t(1) t(2)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial  q^{-7} -2 q^{-6} +4 q^{-5} -5 q^{-4} +6 q^{-3} -5 q^{-2} +2 q+5 q^{-1} -2 (db)
Signature -2 (db)
HOMFLY-PT polynomial a^6 z^2+a^6-a^4 z^4-a^4 z^2+a^4 z^{-2} +a^4-2 a^2 z^4-6 a^2 z^2-2 a^2 z^{-2} -6 a^2+2 z^2+ z^{-2} +4 (db)
Kauffman polynomial z^4 a^8-2 z^2 a^8+2 z^5 a^7-3 z^3 a^7+3 z^6 a^6-7 z^4 a^6+7 z^2 a^6-2 a^6+2 z^7 a^5-3 z^5 a^5+3 z^3 a^5+z^8 a^4-z^6 a^4+3 z^4 a^4-3 z^2 a^4-a^4 z^{-2} +3 a^4+3 z^7 a^3-8 z^5 a^3+13 z^3 a^3-9 z a^3+2 a^3 z^{-1} +z^8 a^2-4 z^6 a^2+14 z^4 a^2-21 z^2 a^2-2 a^2 z^{-2} +11 a^2+z^7 a-3 z^5 a+7 z^3 a-9 z a+2 a z^{-1} +3 z^4-9 z^2- z^{-2} +7 (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=-2 is the signature of L10n95. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L10n95/KhovanovTable
Integral Khovanov Homology

(db, data source)

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

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.