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 L11a305's page at Knotilus.

Visit L11a305's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a305's Link Presentations]

Planar diagram presentation X10,1,11,2 X22,11,9,12 X8,9,1,10 X2,22,3,21 X12,4,13,3 X14,20,15,19 X18,8,19,7 X16,6,17,5 X4,16,5,15 X6,18,7,17 X20,14,21,13
Gauss code {1, -4, 5, -9, 8, -10, 7, -3}, {3, -1, 2, -5, 11, -6, 9, -8, 10, -7, 6, -11, 4, -2}
A Braid Representative
A Morse Link Presentation L11a305 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^3 v^3-3 u^3 v^2+u^3 v-3 u^2 v^3+8 u^2 v^2-5 u^2 v+u^2+u v^3-5 u v^2+8 u v-3 u+v^2-3 v+2}{u^{3/2} v^{3/2}} (db)
Jones polynomial 14 q^{9/2}-15 q^{7/2}+12 q^{5/2}-9 q^{3/2}+\frac{1}{q^{3/2}}-q^{19/2}+3 q^{17/2}-6 q^{15/2}+10 q^{13/2}-13 q^{11/2}+5 \sqrt{q}-\frac{3}{\sqrt{q}} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z^5 a^{-7} -3 z^3 a^{-7} -2 z a^{-7} +z^7 a^{-5} +4 z^5 a^{-5} +5 z^3 a^{-5} +2 z a^{-5} - a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +5 z^3 a^{-3} +3 z a^{-3} + a^{-3} z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -z a^{-1} (db)
Kauffman polynomial z^5 a^{-11} -2 z^3 a^{-11} +z a^{-11} +3 z^6 a^{-10} -6 z^4 a^{-10} +3 z^2 a^{-10} +4 z^7 a^{-9} -5 z^5 a^{-9} -z^3 a^{-9} +z a^{-9} +4 z^8 a^{-8} -4 z^6 a^{-8} -z^2 a^{-8} +3 z^9 a^{-7} -3 z^7 a^{-7} +5 z^5 a^{-7} -7 z^3 a^{-7} +2 z a^{-7} +z^{10} a^{-6} +5 z^8 a^{-6} -15 z^6 a^{-6} +19 z^4 a^{-6} -8 z^2 a^{-6} +6 z^9 a^{-5} -15 z^7 a^{-5} +17 z^5 a^{-5} -6 z^3 a^{-5} -z a^{-5} + a^{-5} z^{-1} +z^{10} a^{-4} +5 z^8 a^{-4} -21 z^6 a^{-4} +25 z^4 a^{-4} -7 z^2 a^{-4} - a^{-4} +3 z^9 a^{-3} -5 z^7 a^{-3} -4 z^5 a^{-3} +10 z^3 a^{-3} -5 z a^{-3} + a^{-3} z^{-1} +4 z^8 a^{-2} -12 z^6 a^{-2} +9 z^4 a^{-2} -2 z^2 a^{-2} +3 z^7 a^{-1} -10 z^5 a^{-1} +8 z^3 a^{-1} -2 z a^{-1} +z^6-3 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 L11a305. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L11a305/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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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