L10n51

From Knot Atlas
Jump to: navigation, search

L10n50.gif

L10n50

L10n52.gif

L10n52

Contents

L10n51.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n51 at Knotilus!


Link Presentations

[edit Notes on L10n51's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X11,19,12,18 X5,12,6,13 X17,4,18,5 X14,7,15,8 X16,14,17,13 X20,15,7,16 X6,19,1,20
Gauss code {1, -2, 3, 6, -5, -10}, {7, -1, 2, -3, -4, 5, 8, -7, 9, -8, -6, 4, 10, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L10n51 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-2 u^2 v^2+u^2 v+u v^4-3 u v^3+3 u v^2-3 u v+u+v^3-2 v^2}{u v^2} (db)
Jones polynomial -\frac{2}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{5}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z+a^9 z^{-1} -2 a^7 z^3-4 a^7 z-2 a^7 z^{-1} +a^5 z^5+3 a^5 z^3+4 a^5 z+2 a^5 z^{-1} -2 a^3 z^3-4 a^3 z-a^3 z^{-1} (db)
Kauffman polynomial -z^5 a^{11}+3 z^3 a^{11}-2 z a^{11}-2 z^6 a^{10}+5 z^4 a^{10}-2 z^2 a^{10}-2 z^7 a^9+4 z^5 a^9-2 z^3 a^9+3 z a^9-a^9 z^{-1} -z^8 a^8+2 z^4 a^8-4 z^7 a^7+11 z^5 a^7-15 z^3 a^7+9 z a^7-2 a^7 z^{-1} -z^8 a^6+z^6 a^6-3 z^4 a^6+2 z^2 a^6-a^6-2 z^7 a^5+6 z^5 a^5-13 z^3 a^5+9 z a^5-2 a^5 z^{-1} -z^6 a^4-3 z^3 a^3+5 z a^3-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).   
\ r
  \  
j \
-8-7-6-5-4-3-2-10χ
-2        22
-4       21-1
-6      31 2
-8     32  -1
-10    33   0
-12   23    1
-14  23     -1
-16 13      2
-18 1       -1
-201        1
Integral Khovanov Homology

(db, data source)

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

L10n50.gif

L10n50

L10n52.gif

L10n52