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 L10n66's page at Knotilus.

Visit L10n66's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n66's Link Presentations]

Planar diagram presentation X6172 X3,11,4,10 X16,12,17,11 X18,13,19,14 X20,18,9,17 X12,19,13,20 X15,8,16,5 X7,14,8,15 X2536 X9,1,10,4
Gauss code {1, -9, -2, 10}, {9, -1, -8, 7}, {-10, 2, 3, -6, 4, 8, -7, -3, 5, -4, 6, -5}
A Braid Representative
A Morse Link Presentation L10n66 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u v w^2+u v w+u w^3-2 u w^2+2 v w-v-w^2+w}{\sqrt{u} \sqrt{v} w^{3/2}} (db)
Jones polynomial  q^{-6} - q^{-5} +2 q^{-4} +q^3- q^{-3} -q^2+ q^{-2} +q+1 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 z^{-2} +a^6-2 a^4 z^2-3 a^4 z^{-2} -5 a^4+a^2 z^4+5 a^2 z^2+4 a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +8 a^2+2 a^{-2} -z^4-5 z^2-3 z^{-2} -6 (db)
Kauffman polynomial a^5 z^7+a^3 z^7+a z^7+z^7 a^{-1} +a^6 z^6+3 a^4 z^6+3 a^2 z^6+z^6 a^{-2} +2 z^6-4 a^5 z^5-5 a^3 z^5-6 a z^5-5 z^5 a^{-1} -5 a^6 z^4-16 a^4 z^4-20 a^2 z^4-5 z^4 a^{-2} -14 z^4+2 a^5 z^3+4 a^3 z^3+6 a z^3+4 z^3 a^{-1} +7 a^6 z^2+23 a^4 z^2+35 a^2 z^2+6 z^2 a^{-2} +25 z^2+a^5 z+a^3 z+a z+z a^{-1} -4 a^6-14 a^4-21 a^2-4 a^{-2} -14-a^5 z^{-1} -a^3 z^{-1} -a z^{-1} - a^{-1} z^{-1} +a^6 z^{-2} +3 a^4 z^{-2} +4 a^2 z^{-2} + a^{-2} z^{-2} +3 z^{-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 L10n66. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7          11
5           0
3        11 0
1      31   2
-1     131   1
-3    122    1
-5   11      0
-7  111      1
-9 12        1
-11           0
-131          1
Integral Khovanov Homology

(db, data source)

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

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