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 L10n97's page at Knotilus.

Visit L10n97's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n97's Link Presentations]

Planar diagram presentation X6172 X2536 X20,13,15,14 X3,11,4,10 X9,1,10,4 X7,17,8,16 X15,5,16,8 X18,11,19,12 X12,19,13,20 X14,17,9,18
Gauss code {1, -2, -4, 5}, {2, -1, -6, 7}, {-5, 4, 8, -9, 3, -10}, {-7, 6, 10, -8, 9, -3}
A Braid Representative
A Morse Link Presentation L10n97 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(4)^2 t(3)^2+t(1) t(4) t(3)^2+t(2) t(4)^2 t(3)+t(1) t(3)-t(1) t(4) t(3)-t(2) t(4) t(3)-t(1) t(2)+t(2) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)} (db)
Jones polynomial -\frac{2}{q^{9/2}}+\frac{1}{q^{7/2}}-q^{5/2}-\frac{3}{q^{5/2}}-\frac{1}{q^{13/2}}+\frac{1}{q^{11/2}}-\sqrt{q}-\frac{2}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^5 z^3+a^5 z^{-3} +3 a^5 z+3 a^5 z^{-1} -a^3 z^5-6 a^3 z^3-3 a^3 z^{-3} -13 a^3 z-10 a^3 z^{-1} +a z^5+7 a z^3+3 a z^{-3} -z^3 a^{-1} - a^{-1} z^{-3} +14 a z+11 a z^{-1} -4 z a^{-1} -4 a^{-1} z^{-1} (db)
Kauffman polynomial -z^5 a^7+4 z^3 a^7-3 z a^7+a^7 z^{-1} -z^6 a^6+3 z^4 a^6-a^6-z^7 a^5+4 z^5 a^5-4 z^3 a^5+3 z a^5-3 a^5 z^{-1} +a^5 z^{-3} -2 z^6 a^4+10 z^4 a^4-16 z^2 a^4-3 a^4 z^{-2} +11 a^4-z^7 a^3+7 z^5 a^3-18 z^3 a^3+21 z a^3-12 a^3 z^{-1} +3 a^3 z^{-3} -2 z^6 a^2+15 z^4 a^2-33 z^2 a^2-6 a^2 z^{-2} +24 a^2-z^7 a+9 z^5 a-25 z^3 a+28 z a-14 a z^{-1} +3 a z^{-3} -z^6+8 z^4-17 z^2-3 z^{-2} +13-z^7 a^{-1} +7 z^5 a^{-1} -15 z^3 a^{-1} +13 z a^{-1} -6 a^{-1} z^{-1} + a^{-1} z^{-3} (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 L10n97. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
6          11
4          11
2        1  1
0      3    3
-2     141   2
-4    3      3
-6   141     2
-8  111      1
-10  1        1
-1211         0
-141          1
Integral Khovanov Homology

(db, data source)

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

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