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)


From Knot Atlas
Jump to: navigation, search






(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n76's page at Knotilus.

Visit L10n76's page at the original Knot Atlas.

Link Presentations

[edit Notes on L10n76's Link Presentations]

Planar diagram presentation X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X4,17,1,18 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 L10n76 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) (w-1) (v+w)}{\sqrt{u} v w} (db)
Jones polynomial  q^{-9} -2 q^{-8} +4 q^{-7} -4 q^{-6} +6 q^{-5} -5 q^{-4} +5 q^{-3} -3 q^{-2} +2 q^{-1} (db)
Signature -2 (db)
HOMFLY-PT polynomial z^2 a^8+a^8 z^{-2} +a^8-z^4 a^6-2 z^2 a^6-2 a^6 z^{-2} -3 a^6-z^4 a^4-z^2 a^4+a^4 z^{-2} +2 z^2 a^2+2 a^2 (db)
Kauffman polynomial a^{10} z^6-4 a^{10} z^4+5 a^{10} z^2-2 a^{10}+2 a^9 z^7-7 a^9 z^5+5 a^9 z^3+a^9 z+a^8 z^8-8 a^8 z^4+6 a^8 z^2-a^8 z^{-2} +a^8+4 a^7 z^7-11 a^7 z^5+8 a^7 z^3-6 a^7 z+2 a^7 z^{-1} +a^6 z^8+a^6 z^6-6 a^6 z^4-2 a^6 z^{-2} +5 a^6+2 a^5 z^7-3 a^5 z^5+4 a^5 z^3-6 a^5 z+2 a^5 z^{-1} +2 a^4 z^6-2 a^4 z^4+2 a^4 z^2-a^4 z^{-2} +a^4+a^3 z^5+a^3 z^3+a^3 z+3 a^2 z^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 L10n76. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
-1        22
-3       32-1
-5      2  2
-7     33  0
-9    32   1
-11   13    2
-13  33     0
-15 13      2
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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.