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 L11a340's page at Knotilus.

Visit L11a340's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a340's Link Presentations]

Planar diagram presentation X10,1,11,2 X14,4,15,3 X22,5,9,6 X6,9,7,10 X20,12,21,11 X18,14,19,13 X12,20,13,19 X16,8,17,7 X4,16,5,15 X8,18,1,17 X2,21,3,22
Gauss code {1, -11, 2, -9, 3, -4, 8, -10}, {4, -1, 5, -7, 6, -2, 9, -8, 10, -6, 7, -5, 11, -3}
A Braid Representative
A Morse Link Presentation L11a340 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(2 u^2 v^2-2 u v^2+7 u v-2 u+2\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial -12 q^{9/2}+16 q^{7/2}-\frac{1}{q^{7/2}}-20 q^{5/2}+\frac{3}{q^{5/2}}+19 q^{3/2}-\frac{8}{q^{3/2}}+q^{15/2}-3 q^{13/2}+7 q^{11/2}-17 \sqrt{q}+\frac{13}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} -z^7 a^{-3} +a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} +3 a z^3-7 z^3 a^{-1} -6 z^3 a^{-3} +3 z^3 a^{-5} +4 a z-5 z a^{-1} -2 z a^{-3} +3 z a^{-5} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 z^{10} a^{-2} -2 z^{10} a^{-4} -5 z^9 a^{-1} -10 z^9 a^{-3} -5 z^9 a^{-5} -7 z^8 a^{-2} -5 z^8 a^{-4} -5 z^8 a^{-6} -7 z^8-6 a z^7+z^7 a^{-1} +21 z^7 a^{-3} +11 z^7 a^{-5} -3 z^7 a^{-7} -3 a^2 z^6+19 z^6 a^{-2} +20 z^6 a^{-4} +13 z^6 a^{-6} -z^6 a^{-8} +10 z^6-a^3 z^5+10 a z^5+12 z^5 a^{-1} -17 z^5 a^{-3} -10 z^5 a^{-5} +8 z^5 a^{-7} +4 a^2 z^4-16 z^4 a^{-2} -20 z^4 a^{-4} -11 z^4 a^{-6} +3 z^4 a^{-8} -6 z^4+2 a^3 z^3-8 a z^3-17 z^3 a^{-1} +6 z^3 a^{-3} +8 z^3 a^{-5} -5 z^3 a^{-7} -a^2 z^2+4 z^2 a^{-2} +7 z^2 a^{-4} +5 z^2 a^{-6} -2 z^2 a^{-8} +3 z^2-a^3 z+5 a z+8 z a^{-1} -2 z a^{-3} -3 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 L11a340. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L11a340/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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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