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)

L10n79

 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n79's page at Knotilus. Visit L10n79's page at the original Knot Atlas.

 Planar diagram presentation X6172 X5,14,6,15 X8493 X2,16,3,15 X16,7,17,8 X9,18,10,19 X11,20,12,13 X13,12,14,5 X4,17,1,18 X19,10,20,11 Gauss code {1, -4, 3, -9}, {-2, -1, 5, -3, -6, 10, -7, 8}, {-8, 2, 4, -5, 9, 6, -10, 7}

Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(u-1) (v-1) (w-1) (v w+1)}{\sqrt{u} v w}$ (db) Jones polynomial $2 q^{-2} -2 q^{-3} +5 q^{-4} -5 q^{-5} +6 q^{-6} -5 q^{-7} +4 q^{-8} -2 q^{-9} + q^{-10}$ (db) Signature -4 (db) HOMFLY-PT polynomial $z^4 a^8+3 z^2 a^8+a^8 z^{-2} +3 a^8-z^6 a^6-5 z^4 a^6-10 z^2 a^6-2 a^6 z^{-2} -9 a^6+2 z^4 a^4+7 z^2 a^4+a^4 z^{-2} +6 a^4$ (db) Kauffman polynomial $a^{12} z^4-2 a^{12} z^2+a^{12}+2 a^{11} z^5-3 a^{11} z^3+2 a^{10} z^6-a^{10} z^4-2 a^{10} z^2+2 a^9 z^7-3 a^9 z^5+3 a^9 z^3+a^8 z^8-a^8 z^6+3 a^8 z^4-3 a^8 z^2-a^8 z^{-2} +3 a^8+3 a^7 z^7-8 a^7 z^5+13 a^7 z^3-9 a^7 z+2 a^7 z^{-1} +a^6 z^8-3 a^6 z^6+8 a^6 z^4-12 a^6 z^2-2 a^6 z^{-2} +9 a^6+a^5 z^7-3 a^5 z^5+7 a^5 z^3-9 a^5 z+2 a^5 z^{-1} +3 a^4 z^4-9 a^4 z^2-a^4 z^{-2} +6 a^4$ (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 $s=$-4 is the signature of L10n79. Nonzero entries off the critical diagonals (if any exist) are highlighted in red. Data:L10n79/KhovanovTable
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-5$ $i=-3$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.