L10n13

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L10n12.gif

L10n12

L10n14.gif

L10n14

Contents

L10n13.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10n13 at Knotilus!


Link Presentations

[edit Notes on L10n13's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X4,19,1,20 X9,14,10,15 X3849 X5,13,6,12 X13,5,14,20 X11,16,12,17 X15,10,16,11 X17,2,18,3
Gauss code {1, 10, -5, -3}, {-6, -1, 2, 5, -4, 9, -8, 6, -7, 4, -9, 8, -10, -2, 3, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L10n13 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^5-2 t(1) t(2)^4+t(1) t(2)^3-t(2)^3-t(1) t(2)^2+t(2)^2-2 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{3}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{3}{q^{5/2}}+\frac{1}{q^{3/2}}-\frac{1}{q^{17/2}}+\frac{2}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{3}{q^{11/2}}-\frac{1}{\sqrt{q}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^9 z+a^9 z^{-1} -a^7 z^5-5 a^7 z^3-7 a^7 z-3 a^7 z^{-1} +a^5 z^7+6 a^5 z^5+12 a^5 z^3+11 a^5 z+4 a^5 z^{-1} -a^3 z^5-5 a^3 z^3-7 a^3 z-2 a^3 z^{-1} (db)
Kauffman polynomial a^{11} z+2 a^{10} z^2-a^{10}+a^9 z^5-a^9 z^3-a^9 z+a^9 z^{-1} +3 a^8 z^6-11 a^8 z^4+10 a^8 z^2-3 a^8+3 a^7 z^7-13 a^7 z^5+17 a^7 z^3-11 a^7 z+3 a^7 z^{-1} +a^6 z^8-a^6 z^6-9 a^6 z^4+12 a^6 z^2-3 a^6+4 a^5 z^7-20 a^5 z^5+30 a^5 z^3-18 a^5 z+4 a^5 z^{-1} +a^4 z^8-4 a^4 z^6+2 a^4 z^4+4 a^4 z^2-2 a^4+a^3 z^7-6 a^3 z^5+12 a^3 z^3-9 a^3 z+2 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 \
-6-5-4-3-2-1012χ
0        11
-2         0
-4      31 2
-6     11  0
-8    22   0
-10  121    0
-12  22     0
-14 12      1
-16 1       -1
-181        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4 i=-2
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} {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}
r=2 {\mathbb Z}_2 {\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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