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 L11a4's page at Knotilus.

Visit L11a4's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a4's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X10,6,11,5 X8493 X22,14,5,13 X20,12,21,11 X12,22,13,21 X14,20,15,19 X18,10,19,9 X2,16,3,15
Gauss code {1, -11, 5, -3}, {4, -1, 2, -5, 10, -4, 7, -8, 6, -9, 11, -2, 3, -10, 9, -7, 8, -6}
A Braid Representative
A Morse Link Presentation L11a4 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(2 t(2)^4-4 t(2)^3+5 t(2)^2-4 t(2)+2\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -q^{19/2}+4 q^{17/2}-9 q^{15/2}+15 q^{13/2}-19 q^{11/2}+22 q^{9/2}-22 q^{7/2}+18 q^{5/2}-14 q^{3/2}+7 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-7} -2 z^3 a^{-7} -z a^{-7} +z^7 a^{-5} +3 z^5 a^{-5} +3 z^3 a^{-5} +2 z a^{-5} + a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +2 z^3 a^{-3} -2 z a^{-3} -3 a^{-3} z^{-1} -z^5 a^{-1} -2 z^3 a^{-1} +z a^{-1} +2 a^{-1} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-4} -2 z^{10} a^{-6} -5 z^9 a^{-3} -11 z^9 a^{-5} -6 z^9 a^{-7} -6 z^8 a^{-2} -8 z^8 a^{-4} -11 z^8 a^{-6} -9 z^8 a^{-8} -4 z^7 a^{-1} +5 z^7 a^{-3} +17 z^7 a^{-5} -8 z^7 a^{-9} +14 z^6 a^{-2} +21 z^6 a^{-4} +21 z^6 a^{-6} +11 z^6 a^{-8} -4 z^6 a^{-10} -z^6+11 z^5 a^{-1} +6 z^5 a^{-3} -10 z^5 a^{-5} +8 z^5 a^{-7} +12 z^5 a^{-9} -z^5 a^{-11} -9 z^4 a^{-2} -13 z^4 a^{-4} -10 z^4 a^{-6} -3 z^4 a^{-8} +5 z^4 a^{-10} +2 z^4-8 z^3 a^{-1} -6 z^3 a^{-3} +9 z^3 a^{-5} -6 z^3 a^{-9} +z^3 a^{-11} +2 z^2 a^{-2} +5 z^2 a^{-4} +5 z^2 a^{-6} -2 z^2 a^{-10} -z a^{-1} -3 z a^{-3} -4 z a^{-5} -2 z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 a^{-3} z^{-1} + a^{-5} 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=3 is the signature of L11a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L11a4/KhovanovTable
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\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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