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)


From Knot Atlas
Jump to: navigation, search






(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11a310's page at Knotilus.

Visit L11a310's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a310's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,4,13,3 X22,12,9,11 X2,9,3,10 X20,17,21,18 X6,14,7,13 X14,8,15,7 X8,16,1,15 X4,20,5,19 X18,6,19,5 X16,21,17,22
Gauss code {1, -4, 2, -9, 10, -6, 7, -8}, {4, -1, 3, -2, 6, -7, 8, -11, 5, -10, 9, -5, 11, -3}
A Braid Representative
A Morse Link Presentation L11a310 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u v+1) (u v-u-2 v+1) (u v-2 u-v+1)}{u^{3/2} v^{3/2}} (db)
Jones polynomial q^{17/2}-3 q^{15/2}+6 q^{13/2}-11 q^{11/2}+14 q^{9/2}-16 q^{7/2}+15 q^{5/2}-14 q^{3/2}+10 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^7 a^{-3} -2 z^5 a^{-1} +4 z^5 a^{-3} -2 z^5 a^{-5} +a z^3-6 z^3 a^{-1} +7 z^3 a^{-3} -6 z^3 a^{-5} +z^3 a^{-7} +2 a z-4 z a^{-1} +7 z a^{-3} -5 z a^{-5} +2 z a^{-7} + a^{-3} z^{-1} - a^{-5} z^{-1} (db)
Kauffman polynomial -z^{10} a^{-2} -z^{10} a^{-4} -3 z^9 a^{-1} -7 z^9 a^{-3} -4 z^9 a^{-5} -7 z^8 a^{-2} -11 z^8 a^{-4} -7 z^8 a^{-6} -3 z^8-a z^7+7 z^7 a^{-1} +13 z^7 a^{-3} -2 z^7 a^{-5} -7 z^7 a^{-7} +32 z^6 a^{-2} +35 z^6 a^{-4} +10 z^6 a^{-6} -5 z^6 a^{-8} +12 z^6+4 a z^5+2 z^5 a^{-1} +7 z^5 a^{-3} +21 z^5 a^{-5} +9 z^5 a^{-7} -3 z^5 a^{-9} -33 z^4 a^{-2} -28 z^4 a^{-4} -5 z^4 a^{-6} +4 z^4 a^{-8} -z^4 a^{-10} -15 z^4-5 a z^3-9 z^3 a^{-1} -16 z^3 a^{-3} -21 z^3 a^{-5} -6 z^3 a^{-7} +3 z^3 a^{-9} +10 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-8} +z^2 a^{-10} +6 z^2+2 a z+3 z a^{-1} +7 z a^{-3} +8 z a^{-5} +z a^{-7} -z a^{-9} + a^{-4} - a^{-3} z^{-1} - a^{-5} 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=3 is the signature of L11a310. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.    Data:L11a310/KhovanovTable
Integral Khovanov Homology

(db, data source)

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

Back to the top.