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 L11a6's page at Knotilus.

Visit L11a6's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a6's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X16,10,17,9 X20,12,21,11 X22,18,5,17 X18,22,19,21 X10,20,11,19 X2,14,3,13
Gauss code {1, -11, 5, -3}, {4, -1, 2, -5, 6, -10, 7, -4, 11, -2, 3, -6, 8, -9, 10, -7, 9, -8}
A Braid Representative
A Morse Link Presentation L11a6 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(2 v^4-4 v^3+5 v^2-4 v+2\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial 22 q^{9/2}-22 q^{7/2}+18 q^{5/2}-15 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+4 q^{17/2}-8 q^{15/2}+14 q^{13/2}-19 q^{11/2}+8 \sqrt{q}-\frac{4}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-7} -2 z^3 a^{-7} + a^{-7} z^{-1} +z^7 a^{-5} +3 z^5 a^{-5} +2 z^3 a^{-5} -2 z a^{-5} -2 a^{-5} z^{-1} +z^7 a^{-3} +3 z^5 a^{-3} +3 z^3 a^{-3} +2 z a^{-3} -z^5 a^{-1} -2 z^3 a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-11} -z^3 a^{-11} +4 z^6 a^{-10} -6 z^4 a^{-10} +2 z^2 a^{-10} +7 z^7 a^{-9} -10 z^5 a^{-9} +4 z^3 a^{-9} +8 z^8 a^{-8} -9 z^6 a^{-8} +5 z^2 a^{-8} -2 a^{-8} +6 z^9 a^{-7} -2 z^7 a^{-7} -6 z^5 a^{-7} +4 z^3 a^{-7} + a^{-7} z^{-1} +2 z^{10} a^{-6} +11 z^8 a^{-6} -22 z^6 a^{-6} +6 z^4 a^{-6} +9 z^2 a^{-6} -5 a^{-6} +11 z^9 a^{-5} -15 z^7 a^{-5} +z^5 a^{-5} +3 z^3 a^{-5} -3 z a^{-5} +2 a^{-5} z^{-1} +2 z^{10} a^{-4} +9 z^8 a^{-4} -21 z^6 a^{-4} +4 z^4 a^{-4} +7 z^2 a^{-4} -3 a^{-4} +5 z^9 a^{-3} -2 z^7 a^{-3} -14 z^5 a^{-3} +12 z^3 a^{-3} -4 z a^{-3} +6 z^8 a^{-2} -11 z^6 a^{-2} +2 z^4 a^{-2} +2 z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} -10 z^5 a^{-1} +8 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} +z^6-2 z^4+z^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=3 is the signature of L11a6. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L11a6/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}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
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}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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