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 L10n53's page at Knotilus.

Visit L10n53's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n53's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X20,13,7,14 X9,15,10,14 X19,11,20,10 X5,16,6,17 X15,18,16,19 X2738 X4,11,5,12 X17,6,18,1
Gauss code {1, -8, 2, -9, -6, 10}, {8, -1, -4, 5, 9, -2, 3, 4, -7, 6, -10, 7, -5, -3}
A Braid Representative
A Morse Link Presentation L10n53 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^2 t(1)^2-3 t(2) t(1)^2+t(1)^2-3 t(2)^2 t(1)+5 t(2) t(1)-3 t(1)+t(2)^2-3 t(2)+1}{t(1) t(2)} (db)
Jones polynomial -\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial 2 a^7 z+a^7 z^{-1} -3 a^5 z^3-5 a^5 z-a^5 z^{-1} +a^3 z^5+2 a^3 z^3+a^3 z-a z^3-a z (db)
Kauffman polynomial 3 a^9 z^3-3 a^9 z+a^8 z^6+2 a^8 z^4-2 a^8 z^2+2 a^7 z^7-2 a^7 z^5+3 a^7 z^3+a^7 z-a^7 z^{-1} +a^6 z^8+2 a^6 z^6-2 a^6 z^4-a^6 z^2+a^6+5 a^5 z^7-8 a^5 z^5+2 a^5 z^3+3 a^5 z-a^5 z^{-1} +a^4 z^8+4 a^4 z^6-11 a^4 z^4+4 a^4 z^2+3 a^3 z^7-5 a^3 z^5+3 a^2 z^6-7 a^2 z^4+3 a^2 z^2+a z^5-2 a z^3+a z (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=-3 is the signature of L10n53. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
2        11
0       2 -2
-2      31 2
-4     43  -1
-6    42   2
-8   34    1
-10  34     -1
-12 13      2
-1413       -2
-162        2
Integral Khovanov Homology

(db, data source)

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