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 L11a50's page at Knotilus.

Visit L11a50's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a50's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X4,19,1,20 X12,6,13,5 X10,4,11,3 X22,14,5,13 X14,22,15,21 X20,12,21,11 X16,9,17,10 X2,16,3,15 X8,17,9,18
Gauss code {1, -10, 5, -3}, {4, -1, 2, -11, 9, -5, 8, -4, 6, -7, 10, -9, 11, -2, 3, -8, 7, -6}
A Braid Representative
A Morse Link Presentation L11a50 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 (u-1) (v-1) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -7 q^{9/2}+\frac{1}{q^{9/2}}+11 q^{7/2}-\frac{4}{q^{7/2}}-15 q^{5/2}+\frac{7}{q^{5/2}}+18 q^{3/2}-\frac{12}{q^{3/2}}-q^{13/2}+3 q^{11/2}-18 \sqrt{q}+\frac{15}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} +z^5 a^{-3} -a^3 z^3+z^3 a^{-3} +2 z a^{-3} +a^3 z^{-1} +2 a^{-3} z^{-1} +a z^5+2 z^5 a^{-1} +3 z^3 a^{-1} -2 a z+z a^{-1} -a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} -2 z^3 a^{-7} +3 z^6 a^{-6} -5 z^4 a^{-6} +6 z^7 a^{-5} -14 z^5 a^{-5} +12 z^3 a^{-5} -5 z a^{-5} + a^{-5} z^{-1} +7 z^8 a^{-4} +a^4 z^6-17 z^6 a^{-4} -2 a^4 z^4+19 z^4 a^{-4} -8 z^2 a^{-4} + a^{-4} +5 z^9 a^{-3} +4 a^3 z^7-7 z^7 a^{-3} -11 a^3 z^5-z^5 a^{-3} +6 a^3 z^3+15 z^3 a^{-3} +a^3 z-11 z a^{-3} -a^3 z^{-1} +2 a^{-3} z^{-1} +2 z^{10} a^{-2} +6 a^2 z^8+5 z^8 a^{-2} -16 a^2 z^6-22 z^6 a^{-2} +10 a^2 z^4+31 z^4 a^{-2} -2 a^2 z^2-15 z^2 a^{-2} +a^2+3 a^{-2} +5 a z^9+10 z^9 a^{-1} -10 a z^7-27 z^7 a^{-1} +4 a z^5+29 z^5 a^{-1} -3 a z^3-8 z^3 a^{-1} +3 a z-4 z a^{-1} -a z^{-1} + a^{-1} z^{-1} +2 z^{10}+4 z^8-19 z^6+19 z^4-9 z^2+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=1 is the signature of L11a50. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L11a50/KhovanovTable
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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).

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