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 L10n69's page at Knotilus.

Visit L10n69's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n69's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X2,14,3,13 X14,7,15,8 X9,18,10,19 X11,16,12,17 X17,20,18,11 X15,1,16,4 X19,10,20,5
Gauss code {1, -4, -3, 9}, {-2, -1, 5, 3, -6, 10}, {-7, 2, 4, -5, -9, 7, -8, 6, -10, 8}
A Braid Representative
A Morse Link Presentation L10n69 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v w^2-2 u v w-u w^2+2 u w-2 v^2 w+v^2+2 v w-2 v}{\sqrt{u} v w} (db)
Jones polynomial  q^{-9} -2 q^{-8} +3 q^{-7} -4 q^{-6} +5 q^{-5} -4 q^{-4} +5 q^{-3} -2 q^{-2} +2 q^{-1} (db)
Signature -2 (db)
HOMFLY-PT polynomial z^2 a^8+a^8-z^4 a^6-2 z^2 a^6+a^6 z^{-2} -a^6-z^4 a^4-2 z^2 a^4-2 a^4 z^{-2} -3 a^4+2 z^2 a^2+a^2 z^{-2} +3 a^2 (db)
Kauffman polynomial a^{10} z^6-4 a^{10} z^4+4 a^{10} z^2-a^{10}+2 a^9 z^7-8 a^9 z^5+8 a^9 z^3-2 a^9 z+a^8 z^8-a^8 z^6-6 a^8 z^4+6 a^8 z^2-a^8+4 a^7 z^7-14 a^7 z^5+14 a^7 z^3-6 a^7 z+a^6 z^8-7 a^6 z^4+8 a^6 z^2+a^6 z^{-2} -3 a^6+2 a^5 z^7-5 a^5 z^5+5 a^5 z^3-2 a^5 z^{-1} +2 a^4 z^6-5 a^4 z^4+9 a^4 z^2+2 a^4 z^{-2} -6 a^4+a^3 z^5-a^3 z^3+4 a^3 z-2 a^3 z^{-1} +3 a^2 z^2+a^2 z^{-2} -4 a^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=-2 is the signature of L10n69. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-1        22
-3       220
-5      3  3
-7     23  1
-9    32   1
-11   12    1
-13  23     -1
-15 12      1
-17 1       -1
-191        1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{2} {\mathbb Z}^{2}

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.