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 L11a302's page at Knotilus.

Visit L11a302's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a302's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X4,9,5,10 X16,6,17,5 X22,14,9,13 X20,16,21,15 X14,22,15,21 X18,8,19,7 X6,18,7,17 X8,20,1,19
Gauss code {1, -2, 3, -4, 5, -10, 9, -11}, {4, -1, 2, -3, 6, -8, 7, -5, 10, -9, 11, -7, 8, -6}
A Braid Representative
A Morse Link Presentation L11a302 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 (u-1) (v-1) (u v+1)^2}{u^{3/2} v^{3/2}} (db)
Jones polynomial -7 q^{9/2}+8 q^{7/2}-\frac{1}{q^{7/2}}-10 q^{5/2}+\frac{2}{q^{5/2}}+10 q^{3/2}-\frac{4}{q^{3/2}}+q^{15/2}-2 q^{13/2}+4 q^{11/2}-9 \sqrt{q}+\frac{6}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} -z^7 a^{-3} +a z^5-5 z^5 a^{-1} -5 z^5 a^{-3} +z^5 a^{-5} +4 a z^3-8 z^3 a^{-1} -8 z^3 a^{-3} +4 z^3 a^{-5} +4 a z-4 z a^{-1} -4 z a^{-3} +4 z a^{-5} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-2} -z^{10} a^{-4} -2 z^9 a^{-1} -4 z^9 a^{-3} -2 z^9 a^{-5} +z^8 a^{-2} +2 z^8 a^{-4} -2 z^8 a^{-6} -3 z^8-3 a z^7+4 z^7 a^{-1} +16 z^7 a^{-3} +7 z^7 a^{-5} -2 z^7 a^{-7} -2 a^2 z^6+2 z^6 a^{-2} -3 z^6 a^{-4} +5 z^6 a^{-6} -z^6 a^{-8} +9 z^6-a^3 z^5+9 a z^5+z^5 a^{-1} -30 z^5 a^{-3} -14 z^5 a^{-5} +7 z^5 a^{-7} +5 a^2 z^4-7 z^4 a^{-2} +2 z^4 a^{-4} -2 z^4 a^{-6} +4 z^4 a^{-8} -10 z^4+3 a^3 z^3-9 a z^3-13 z^3 a^{-1} +23 z^3 a^{-3} +18 z^3 a^{-5} -6 z^3 a^{-7} -a^2 z^2+2 z^2 a^{-4} +z^2 a^{-6} -4 z^2 a^{-8} +2 z^2-a^3 z+5 a z+6 z a^{-1} -6 z a^{-3} -5 z a^{-5} +z a^{-7} +1-a z^{-1} - a^{-1} 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=1 is the signature of L11a302. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L11a302/KhovanovTable
Integral Khovanov Homology

(db, data source)

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