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 L11a347's page at Knotilus.

Visit L11a347's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a347's Link Presentations]

Planar diagram presentation X12,1,13,2 X8493 X20,14,21,13 X22,16,11,15 X14,22,15,21 X6,18,7,17 X18,8,19,7 X16,20,17,19 X10,6,1,5 X4,10,5,9 X2,11,3,12
Gauss code {1, -11, 2, -10, 9, -6, 7, -2, 10, -9}, {11, -1, 3, -5, 4, -8, 6, -7, 8, -3, 5, -4}
A Braid Representative
A Morse Link Presentation L11a347 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2+2 t(1)^2-t(2)^2 t(1)-t(1)+2 t(2)^2-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -12 q^{9/2}+7 q^{7/2}-5 q^{5/2}+q^{3/2}+q^{23/2}-4 q^{21/2}+8 q^{19/2}-12 q^{17/2}+15 q^{15/2}-15 q^{13/2}+15 q^{11/2}-\sqrt{q} (db)
Signature 5 (db)
HOMFLY-PT polynomial -z^7 a^{-5} -z^7 a^{-7} +z^5 a^{-3} -5 z^5 a^{-5} -3 z^5 a^{-7} +z^5 a^{-9} +5 z^3 a^{-3} -11 z^3 a^{-5} +2 z^3 a^{-9} +8 z a^{-3} -14 z a^{-5} +6 z a^{-7} +4 a^{-3} z^{-1} -8 a^{-5} z^{-1} +5 a^{-7} z^{-1} - a^{-9} z^{-1} (db)
Kauffman polynomial z^4 a^{-14} +4 z^5 a^{-13} -2 z^3 a^{-13} +8 z^6 a^{-12} -8 z^4 a^{-12} +2 z^2 a^{-12} +10 z^7 a^{-11} -13 z^5 a^{-11} +4 z^3 a^{-11} +z a^{-11} +8 z^8 a^{-10} -9 z^6 a^{-10} +3 z^2 a^{-10} -2 a^{-10} +4 z^9 a^{-9} -5 z^5 a^{-9} -5 z^3 a^{-9} +3 z a^{-9} + a^{-9} z^{-1} +z^{10} a^{-8} +6 z^8 a^{-8} -10 z^6 a^{-8} -11 z^4 a^{-8} +20 z^2 a^{-8} -9 a^{-8} +5 z^9 a^{-7} -10 z^7 a^{-7} +3 z^5 a^{-7} +3 z^3 a^{-7} -7 z a^{-7} +5 a^{-7} z^{-1} +z^{10} a^{-6} -z^8 a^{-6} +5 z^6 a^{-6} -25 z^4 a^{-6} +33 z^2 a^{-6} -14 a^{-6} +z^9 a^{-5} +z^7 a^{-5} -15 z^5 a^{-5} +27 z^3 a^{-5} -21 z a^{-5} +8 a^{-5} z^{-1} +z^8 a^{-4} -2 z^6 a^{-4} -5 z^4 a^{-4} +14 z^2 a^{-4} -8 a^{-4} +z^7 a^{-3} -6 z^5 a^{-3} +13 z^3 a^{-3} -12 z a^{-3} +4 a^{-3} 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=5 is the signature of L11a347. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L11a347/KhovanovTable
Integral Khovanov Homology

(db, data source)

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