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 L10n54's page at Knotilus.

Visit L10n54's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n54's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X20,12,9,11 X2,9,3,10 X17,5,18,4 X5,19,6,18 X6,14,7,13 X14,8,15,7 X8,16,1,15 X19,17,20,16
Gauss code {1, -4, 2, 5, -6, -7, 8, -9}, {4, -1, 3, -2, 7, -8, 9, 10, -5, 6, -10, -3}
A Braid Representative
A Morse Link Presentation L10n54 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^3 t(1)^3-t(2)^2 t(1)^3-t(2)^3 t(1)^2-t(1)-t(2)+1}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -2 q^{9/2}+2 q^{7/2}-2 q^{5/2}+q^{3/2}+q^{15/2}-2 q^{13/2}+q^{11/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial -z a^{-9} +z^5 a^{-7} +5 z^3 a^{-7} +5 z a^{-7} -z^7 a^{-5} -6 z^5 a^{-5} -11 z^3 a^{-5} -8 z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} +5 z^3 a^{-3} +6 z a^{-3} + a^{-3} z^{-1} (db)
Kauffman polynomial z^6 a^{-8} -4 z^4 a^{-8} +2 z^2 a^{-8} +2 z^7 a^{-7} -10 z^5 a^{-7} +12 z^3 a^{-7} -3 z a^{-7} +z^8 a^{-6} -4 z^6 a^{-6} +2 z^4 a^{-6} +z^2 a^{-6} +3 z^7 a^{-5} -16 z^5 a^{-5} +23 z^3 a^{-5} -10 z a^{-5} + a^{-5} z^{-1} +z^8 a^{-4} -5 z^6 a^{-4} +6 z^4 a^{-4} -z^2 a^{-4} - a^{-4} +z^7 a^{-3} -6 z^5 a^{-3} +11 z^3 a^{-3} -7 z a^{-3} + a^{-3} z^{-1} (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=5 is the signature of L10n54. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
16       1-1
14      1 1
12     12 1
10    111 1
8   11   0
6  11    0
4 12     1
2        0
01       1
Integral Khovanov Homology

(db, data source)

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