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 L10n58's page at Knotilus.

Visit L10n58's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n58's Link Presentations]

Planar diagram presentation X12,1,13,2 X7,17,8,16 X5,1,6,10 X3746 X9,5,10,4 X17,11,18,20 X13,19,14,18 X19,15,20,14 X2,11,3,12 X15,9,16,8
Gauss code {1, -9, -4, 5, -3, 4, -2, 10, -5, 3}, {9, -1, -7, 8, -10, 2, -6, 7, -8, 6}
A Braid Representative
A Morse Link Presentation L10n58 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(u^2 v+u v^2-2 u v+u+v\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial 8 q^{9/2}-8 q^{7/2}+4 q^{5/2}-3 q^{3/2}-q^{19/2}+3 q^{17/2}-5 q^{15/2}+8 q^{13/2}-8 q^{11/2} (db)
Signature 3 (db)
HOMFLY-PT polynomial -z a^{-9} - a^{-9} z^{-1} +3 z^3 a^{-7} +7 z a^{-7} +5 a^{-7} z^{-1} -2 z^5 a^{-5} -8 z^3 a^{-5} -13 z a^{-5} -8 a^{-5} z^{-1} +3 z^3 a^{-3} +7 z a^{-3} +4 a^{-3} z^{-1} (db)
Kauffman polynomial z^5 a^{-11} -2 z^3 a^{-11} +z a^{-11} +3 z^6 a^{-10} -7 z^4 a^{-10} +5 z^2 a^{-10} -2 a^{-10} +3 z^7 a^{-9} -3 z^5 a^{-9} -4 z^3 a^{-9} +z a^{-9} + a^{-9} z^{-1} +z^8 a^{-8} +8 z^6 a^{-8} -26 z^4 a^{-8} +24 z^2 a^{-8} -9 a^{-8} +7 z^7 a^{-7} -14 z^5 a^{-7} +12 z^3 a^{-7} -9 z a^{-7} +5 a^{-7} z^{-1} +z^8 a^{-6} +8 z^6 a^{-6} -25 z^4 a^{-6} +31 z^2 a^{-6} -14 a^{-6} +4 z^7 a^{-5} -10 z^5 a^{-5} +20 z^3 a^{-5} -19 z a^{-5} +8 a^{-5} z^{-1} +3 z^6 a^{-4} -6 z^4 a^{-4} +12 z^2 a^{-4} -8 a^{-4} +6 z^3 a^{-3} -10 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=3 is the signature of L10n58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
20        11
18       2 -2
16      31 2
14     52  -3
12    33   0
10   55    0
8  33     0
6 15      4
423       -1
23        3
Integral Khovanov Homology

(db, data source)

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

Back to the top.