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 L10n63's page at Knotilus.

Visit L10n63's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n63's Link Presentations]

Planar diagram presentation X12,1,13,2 X7,17,8,16 X3948 X17,2,18,3 X5,14,6,15 X11,6,12,7 X9,18,10,19 X15,11,16,20 X10,13,1,14 X19,5,20,4
Gauss code {1, 4, -3, 10, -5, 6, -2, 3, -7, -9}, {-6, -1, 9, 5, -8, 2, -4, 7, -10, 8}
A Braid Representative
A Morse Link Presentation L10n63 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u v+1) \left(u^2 v^2-2 u^2 v+u^2-2 u v^2+3 u v-2 u+v^2-2 v+1\right)}{u^{3/2} v^{3/2}} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-7 q^{3/2}+9 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{10}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{3}{q^{9/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z+a^5 z^{-1} -a^3 z^5-3 a^3 z^3-3 a^3 z-a^3 z^{-1} +a z^7+4 a z^5-z^5 a^{-1} +4 a z^3-2 z^3 a^{-1} (db)
Kauffman polynomial -3 a^2 z^8-3 z^8-6 a^3 z^7-12 a z^7-6 z^7 a^{-1} -3 a^4 z^6-a^2 z^6-4 z^6 a^{-2} -2 z^6+13 a^3 z^5+26 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +6 a^2 z^4+7 z^4 a^{-2} +13 z^4-6 a^5 z^3-18 a^3 z^3-18 a z^3-5 z^3 a^{-1} +z^3 a^{-3} -3 a^2 z^2-2 z^2 a^{-2} -5 z^2+6 a^5 z+9 a^3 z+4 a z+z a^{-1} +a^4-a^5 z^{-1} -a^3 z^{-1} (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 L10n63. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
8        11
6       3 -3
4      41 3
2     53  -2
0    64   2
-2   56    1
-4  45     -1
-6 25      3
-814       -3
-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}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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