L10n11

From Knot Atlas
Jump to: navigation, search

L10n10.gif

L10n10

L10n12.gif

L10n12

Contents

L10n11.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n11 at Knotilus!


Link Presentations

[edit Notes on L10n11's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X15,1,16,4 X5,12,6,13 X3849 X9,16,10,17 X17,20,18,5 X11,19,12,18 X19,11,20,10 X2,14,3,13
Gauss code {1, -10, -5, 3}, {-4, -1, 2, 5, -6, 9, -8, 4, 10, -2, -3, 6, -7, 8, -9, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation L10n11 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1)^3}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -2 q^{3/2}+3 \sqrt{q}-\frac{5}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial z^3 a^5+2 z a^5+a^5 z^{-1} -z^5 a^3-4 z^3 a^3-7 z a^3-3 a^3 z^{-1} +3 z^3 a+7 z a+4 a z^{-1} -2 z a^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial a^7 z^5-3 a^7 z^3+a^7 z+2 a^6 z^6-6 a^6 z^4+4 a^6 z^2-a^6+2 a^5 z^7-5 a^5 z^5+3 a^5 z^3-2 a^5 z+a^5 z^{-1} +a^4 z^8-6 a^4 z^4+9 a^4 z^2-3 a^4+4 a^3 z^7-13 a^3 z^5+20 a^3 z^3-13 a^3 z+3 a^3 z^{-1} +a^2 z^8-a^2 z^6+5 a^2 z^2-3 a^2+2 a z^7-7 a z^5+17 a z^3+3 z^3 a^{-1} -15 a z-5 z a^{-1} +4 a z^{-1} +2 a^{-1} z^{-1} +z^6-2 (db)

Khovanov Homology

The coefficients of the monomials t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r).   
\ r
  \  
j \
-6-5-4-3-2-1012χ
4        22
2       1 -1
0      42 2
-2     33  0
-4    32   1
-6   23    1
-8  13     -2
-10 12      1
-12 1       -1
-141        1
Integral Khovanov Homology

(db, data source)

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

L10n10.gif

L10n10

L10n12.gif

L10n12