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 L10n75's page at Knotilus.

Visit L10n75's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n75's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X3849 X2,16,3,15 X16,7,17,8 X9,18,10,19 X17,1,18,4 X19,12,20,5 X11,20,12,13 X13,10,14,11
Gauss code {1, -4, -3, 7}, {-2, -1, 5, 3, -6, 10, -9, 8}, {-10, 2, 4, -5, -7, 6, -8, 9}
A Braid Representative
A Morse Link Presentation L10n75 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u v^2 w^2+u v^2 w-u v w^3+2 u v w^2-u v w-u w^2+v^3 w+v^2 w^2-2 v^2 w+v^2-v w^2+v w}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial 2 q^{-1} -3 q^{-2} +5 q^{-3} -4 q^{-4} +5 q^{-5} -4 q^{-6} +3 q^{-7} - q^{-8} + q^{-9} (db)
Signature -2 (db)
HOMFLY-PT polynomial z^2 a^8+a^8 z^{-2} +2 a^8-z^4 a^6-3 z^2 a^6-2 a^6 z^{-2} -5 a^6-z^4 a^4-z^2 a^4+a^4 z^{-2} +a^4+2 z^2 a^2+2 a^2 (db)
Kauffman polynomial z^6 a^{10}-5 z^4 a^{10}+7 z^2 a^{10}-2 a^{10}+z^7 a^9-3 z^5 a^9+z^3 a^9+z a^9+z^8 a^8-3 z^6 a^8+3 z^4 a^8-4 z^2 a^8-a^8 z^{-2} +3 a^8+3 z^7 a^7-9 z^5 a^7+9 z^3 a^7-8 z a^7+2 a^7 z^{-1} +z^8 a^6-2 z^6 a^6+6 z^4 a^6-14 z^2 a^6-2 a^6 z^{-2} +9 a^6+2 z^7 a^5-5 z^5 a^5+9 z^3 a^5-8 z a^5+2 a^5 z^{-1} +2 z^6 a^4-2 z^4 a^4-a^4 z^{-2} +3 a^4+z^5 a^3+z^3 a^3+z a^3+3 z^2 a^2-2 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 L10n75. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-1        22
-3       32-1
-5      2  2
-7     23  1
-9    32   1
-11   23    1
-13  12     -1
-15  2      2
-1711       0
-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} {\mathbb Z}
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\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}^{2}
r=-1 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
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.