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 L10n71's page at Knotilus.

Visit L10n71's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n71's Link Presentations]

Planar diagram presentation X6172 X3,12,4,13 X7,17,8,16 X9,11,10,20 X11,18,12,19 X15,9,16,8 X19,5,20,10 X17,14,18,15 X2536 X13,4,14,1
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 L10n71 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^2 t(2)^2+t(1) t(2)^2-2 t(1) t(3) t(2)^2-t(2)^2-2 t(1) t(3)^2 t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+2 t(2)+t(1) t(3)^2-t(3)^2+2 t(3)-1}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial q^3-2 q^2+5 q-4+7 q^{-1} -6 q^{-2} +5 q^{-3} -4 q^{-4} +2 q^{-5} (db)
Signature -2 (db)
HOMFLY-PT polynomial a^6-a^4 z^4-3 a^4 z^2-2 a^4+a^2 z^6+4 a^2 z^4+5 a^2 z^2+a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +3 a^2+2 a^{-2} -2 z^4-6 z^2-2 z^{-2} -4 (db)
Kauffman polynomial a^2 z^8+z^8+3 a^3 z^7+5 a z^7+2 z^7 a^{-1} +3 a^4 z^6+4 a^2 z^6+z^6 a^{-2} +2 z^6+a^5 z^5-6 a^3 z^5-13 a z^5-6 z^5 a^{-1} -5 a^4 z^4-17 a^2 z^4-4 z^4 a^{-2} -16 z^4+3 a^5 z^3+7 a^3 z^3+6 a z^3+2 z^3 a^{-1} +3 a^6 z^2+6 a^4 z^2+14 a^2 z^2+6 z^2 a^{-2} +17 z^2-2 a^5 z-6 a^3 z+4 z a^{-1} -a^6-a^4-3 a^2-4 a^{-2} -6-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 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=-2 is the signature of L10n71. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7        11
5       1 -1
3      41 3
1     23  1
-1    52   3
-3   23    1
-5  34     -1
-7 12      1
-913       -2
-112        2
Integral Khovanov Homology

(db, data source)

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