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)

L10n88

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 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n88's page at Knotilus. Visit L10n88's page at the original Knot Atlas.

Link Presentations

 Planar diagram presentation X6172 X10,3,11,4 X7,14,8,15 X15,17,16,20 X11,19,12,18 X17,13,18,12 X19,5,20,16 X13,8,14,9 X2536 X4,9,1,10 Gauss code {1, -9, 2, -10}, {-6, 5, -7, 4}, {9, -1, -3, 8, 10, -2, -5, 6, -8, 3, -4, 7}
A Braid Representative
A Morse Link Presentation

Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(3)-1) \left(-t(1) t(3)^2+t(2) t(3)^2-t(3)^2-t(1) t(3)-t(2) t(3)+t(1)-t(1) t(2)-t(2)\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $q^3+3-2 q^{-1} +3 q^{-2} -3 q^{-3} +3 q^{-4} -2 q^{-5} + q^{-6}$ (db) Signature 0 (db) HOMFLY-PT polynomial $a^6-2 z^2 a^4-2 a^4+z^4 a^2+3 z^2 a^2+a^2 z^{-2} +4 a^2-z^4-6 z^2-2 z^{-2} -6+z^2 a^{-2} + a^{-2} z^{-2} +3 a^{-2}$ (db) Kauffman polynomial $a^4 z^8+a^2 z^8+2 a^5 z^7+3 a^3 z^7+a z^7+a^6 z^6-2 a^4 z^6-4 a^2 z^6+z^6 a^{-2} -8 a^5 z^5-13 a^3 z^5-5 a z^5-4 a^6 z^4-4 a^4 z^4+3 a^2 z^4-6 z^4 a^{-2} -3 z^4+7 a^5 z^3+15 a^3 z^3+5 a z^3-3 z^3 a^{-1} +4 a^6 z^2+4 a^4 z^2+a^2 z^2+9 z^2 a^{-2} +10 z^2-2 a^5 z-6 a^3 z+2 a z+6 z a^{-1} -a^6-2 a^2-5 a^{-2} -7-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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=$0 is the signature of L10n88. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-101234χ
7          11
5          11
3       11  0
1      3    3
-1     241   1
-3    21     1
-5   121     0
-7  22       0
-9 12        1
-11 1         -1
-131          1
Integral Khovanov Homology
 $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $i=1$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ $r=4$ ${\mathbb Z}$ ${\mathbb Z}$

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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