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 L10n99's page at Knotilus.

Visit L10n99's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n99's Link Presentations]

Planar diagram presentation X6172 X3,11,4,10 X7,15,8,14 X13,5,14,8 X11,18,12,19 X15,20,16,17 X19,16,20,9 X17,12,18,13 X2536 X9,1,10,4
Gauss code {1, -9, -2, 10}, {9, -1, -3, 4}, {-8, 5, -7, 6}, {-10, 2, -5, 8, -4, 3, -6, 7}
A Braid Representative
A Morse Link Presentation L10n99 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(x-1) \left(u v w x-u v w-2 u v x-u w x+u x-v w x+v x+2 w x+x^2-x\right)}{\sqrt{u} \sqrt{v} \sqrt{w} x^{3/2}} (db)
Jones polynomial -q^{7/2}+2 q^{5/2}-6 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{7}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{3}{q^{9/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^{-3} +2 a^5 z^{-1} +a^3 z^3-3 a^3 z^{-3} -3 a^3 z-6 a^3 z^{-1} -z a^{-3} - a^{-3} z^{-1} -a z^5-a z^3+3 a z^{-3} +2 z^3 a^{-1} - a^{-1} z^{-3} +2 a z+5 a z^{-1} +2 z a^{-1} (db)
Kauffman polynomial 6 a^5 z^3-a^5 z^{-3} -10 a^5 z+6 a^5 z^{-1} +3 a^4 z^6-6 a^4 z^4+15 a^4 z^2+3 a^4 z^{-2} -13 a^4+4 a^3 z^7-9 a^3 z^5+z^5 a^{-3} +19 a^3 z^3-3 z^3 a^{-3} -3 a^3 z^{-3} -23 a^3 z+3 z a^{-3} +14 a^3 z^{-1} - a^{-3} z^{-1} +a^2 z^8+8 a^2 z^6+2 z^6 a^{-2} -24 a^2 z^4-3 z^4 a^{-2} +33 a^2 z^2+6 a^2 z^{-2} -24 a^2+ a^{-2} +7 a z^7+3 z^7 a^{-1} -15 a z^5-5 z^5 a^{-1} +19 a z^3+3 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -20 a z-4 z a^{-1} +12 a z^{-1} +3 a^{-1} z^{-1} +z^8+7 z^6-21 z^4+18 z^2+3 z^{-2} -11 (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 L10n99. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
8        11
6       21-1
4      4  4
2     32  -1
0    64   2
-2   57    2
-4  42     2
-6  5      5
-834       -1
-103        3
Integral Khovanov Homology

(db, data source)

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