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 L10n85's page at Knotilus.

Visit L10n85's page at the original Knot Atlas.

Link L10n85.
A graph, L10n85.
A part of a link and a part of a graph.

Link Presentations

[edit Notes on L10n85's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X20,14,15,13 X7,16,8,17 X15,8,16,9 X18,12,19,11 X12,20,13,19 X14,18,5,17 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {-5, 4, 8, -6, 7, -3}, {9, -1, -4, 5, 10, -2, 6, -7, 3, -8}
A Braid Representative
A Morse Link Presentation L10n85 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(w-1) \left(u v^2 w-u v w+2 u v-u-v^2 w+2 v w-v+1\right)}{\sqrt{u} v w} (db)
Jones polynomial q^4+2 q^{-4} -3 q^3-3 q^{-3} +5 q^2+6 q^{-2} -6 q-6 q^{-1} +8 (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^6+a^2 z^4+z^4 a^{-2} -4 z^4+a^2 z^2+2 z^2 a^{-2} -5 z^2+a^4-2 a^2+ a^{-2} +a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial 3 a^4 z^4+z^4 a^{-4} -7 a^4 z^2-z^2 a^{-4} -a^4 z^{-2} +4 a^4+a^3 z^7-a^3 z^5+3 z^5 a^{-3} +2 a^3 z^3-4 z^3 a^{-3} -5 a^3 z+2 a^3 z^{-1} +a^2 z^8-a^2 z^6+4 z^6 a^{-2} +4 a^2 z^4-6 z^4 a^{-2} -8 a^2 z^2+3 z^2 a^{-2} -2 a^2 z^{-2} +6 a^2- a^{-2} +4 a z^7+3 z^7 a^{-1} -7 a z^5-3 z^5 a^{-1} +7 a z^3+z^3 a^{-1} -5 a z+2 a z^{-1} +z^8+3 z^6-6 z^4+3 z^2- z^{-2} +2 (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=0 is the signature of L10n85. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9        11
7       2 -2
5      31 2
3     43  -1
1    42   2
-1   35    2
-3  33     0
-5 14      3
-712       -1
-92        2
Integral Khovanov Homology

(db, data source)

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