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 L10n72's page at Knotilus.

Visit L10n72's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n72's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X7,17,8,16 X9,11,10,20 X11,18,12,19 X15,9,16,8 X19,5,20,10 X17,14,18,15 X2536 X4,14,1,13
Gauss code {1, -9, 2, -10}, {9, -1, -3, 6, -4, 7}, {-5, -2, 10, 8, -6, 3, -8, 5, -7, 4}
A Braid Representative
A Morse Link Presentation L10n72 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^2+u v w^2-u v w-u w^2+u w-v^2 w+v^2+v w-v-w^2}{\sqrt{u} v w} (db)
Jones polynomial q^6+2 q^3-q^2+3 q-2+2 q^{-1} -2 q^{-2} + q^{-3} (db)
Signature 2 (db)
HOMFLY-PT polynomial  a^{-6} z^{-2} + a^{-6} -z^2 a^{-4} -2 a^{-4} z^{-2} -4 a^{-4} +a^2 z^2+2 z^2 a^{-2} + a^{-2} z^{-2} +a^2+4 a^{-2} -z^4-3 z^2-2 (db)
Kauffman polynomial z^8 a^{-2} +z^8+2 a z^7+3 z^7 a^{-1} +z^7 a^{-3} +a^2 z^6-5 z^6 a^{-2} +z^6 a^{-6} -3 z^6-9 a z^5-15 z^5 a^{-1} -6 z^5 a^{-3} -4 a^2 z^4+6 z^4 a^{-2} -2 z^4 a^{-4} -6 z^4 a^{-6} -2 z^4+9 a z^3+19 z^3 a^{-1} +8 z^3 a^{-3} -2 z^3 a^{-5} +3 a^2 z^2-z^2 a^{-2} +6 z^2 a^{-4} +9 z^2 a^{-6} +5 z^2-2 a z-6 z a^{-1} +4 z a^{-5} -a^2-3 a^{-2} -6 a^{-4} -4 a^{-6} -1-2 a^{-3} z^{-1} -2 a^{-5} z^{-1} + a^{-2} z^{-2} +2 a^{-4} z^{-2} + a^{-6} 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=2 is the signature of L10n72. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
13          11
11          11
9       11  0
7      2    2
5     131   1
3    31     2
1   131     1
-1  121      0
-3 11        0
-5 1         -1
-71          1
Integral Khovanov Homology

(db, data source)

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