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 L11a47's page at Knotilus.

Visit L11a47's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a47's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X18,9,19,10 X8,17,9,18 X4,19,1,20 X12,6,13,5 X10,4,11,3 X22,14,5,13 X14,22,15,21 X20,12,21,11 X2,16,3,15
Gauss code {1, -11, 7, -5}, {6, -1, 2, -4, 3, -7, 10, -6, 8, -9, 11, -2, 4, -3, 5, -10, 9, -8}
A Braid Representative
A Morse Link Presentation L11a47 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (v-1) \left(v^2+1\right) \left(2 v^2-3 v+2\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial 3 q^{9/2}-\frac{4}{q^{9/2}}-6 q^{7/2}+\frac{8}{q^{7/2}}+10 q^{5/2}-\frac{13}{q^{5/2}}-15 q^{3/2}+\frac{16}{q^{3/2}}-q^{11/2}+\frac{1}{q^{11/2}}+17 \sqrt{q}-\frac{18}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-z^5 a^{-3} -2 a^3 z^3-3 z^3 a^{-3} -2 z a^{-3} +a^3 z^{-1} - a^{-3} z^{-1} +a z^7+z^7 a^{-1} +3 a z^5+4 z^5 a^{-1} +a z^3+6 z^3 a^{-1} -3 a z+5 z a^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} (db)
Kauffman polynomial a^6 z^4+z^7 a^{-5} +4 a^5 z^5-4 z^5 a^{-5} -2 a^5 z^3+4 z^3 a^{-5} +3 z^8 a^{-4} +8 a^4 z^6-12 z^6 a^{-4} -7 a^4 z^4+14 z^4 a^{-4} +a^4 z^2-4 z^2 a^{-4} +4 z^9 a^{-3} +11 a^3 z^7-15 z^7 a^{-3} -15 a^3 z^5+20 z^5 a^{-3} +6 a^3 z^3-14 z^3 a^{-3} +a^3 z+5 z a^{-3} -a^3 z^{-1} - a^{-3} z^{-1} +2 z^{10} a^{-2} +10 a^2 z^8-14 a^2 z^6-14 z^6 a^{-2} +4 a^2 z^4+16 z^4 a^{-2} +a^2 z^2-5 z^2 a^{-2} +6 a z^9+10 z^9 a^{-1} -4 a z^7-31 z^7 a^{-1} -6 a z^5+37 z^5 a^{-1} -3 a z^3-29 z^3 a^{-1} +8 a z+12 z a^{-1} -2 a z^{-1} -2 a^{-1} z^{-1} +2 z^{10}+7 z^8-24 z^6+14 z^4-z^2-1 (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 L11a47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
12           11
10          2 -2
8         41 3
6        62  -4
4       94   5
2      86    -2
0     109     1
-2    810      2
-4   58       -3
-6  38        5
-8 15         -4
-10 3          3
-121           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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.