L11n10

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L11n9.gif

L11n9

L11n11.gif

L11n11

Contents

L11n10.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n10 at Knotilus!


Link Presentations

[edit Notes on L11n10's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X5,10,6,11 X3849 X13,22,14,5 X11,20,12,21 X21,12,22,13 X19,14,20,15 X9,18,10,19 X15,2,16,3
Gauss code {1, 11, -5, -3}, {-4, -1, 2, 5, -10, 4, -7, 8, -6, 9, -11, -2, 3, 10, -9, 7, -8, 6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n10 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(2)^5-2 t(1) t(2)^4+2 t(1) t(2)^3-t(2)^3-t(1) t(2)^2+2 t(2)^2-2 t(2)+2}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{5}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{4}{q^{21/2}}-\frac{2}{q^{23/2}}+\frac{1}{q^{25/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial a^{13} (-z)-a^{13} z^{-1} +a^{11} z^5+5 a^{11} z^3+6 a^{11} z+a^{11} z^{-1} -a^9 z^7-5 a^9 z^5-6 a^9 z^3+2 a^9 z^{-1} -a^7 z^7-6 a^7 z^5-11 a^7 z^3-7 a^7 z-2 a^7 z^{-1} (db)
Kauffman polynomial a^{16} z^2-a^{16}+2 a^{15} z^3-a^{15} z+a^{14} z^6-2 a^{14} z^4+3 a^{14} z^2+2 a^{13} z^7-7 a^{13} z^5+8 a^{13} z^3-a^{13} z-a^{13} z^{-1} +2 a^{12} z^8-8 a^{12} z^6+11 a^{12} z^4-8 a^{12} z^2+3 a^{12}+a^{11} z^9-3 a^{11} z^7+2 a^{11} z^5-3 a^{11} z^3+2 a^{11} z-a^{11} z^{-1} +3 a^{10} z^8-13 a^{10} z^6+15 a^{10} z^4-6 a^{10} z^2+a^9 z^9-4 a^9 z^7+3 a^9 z^5+2 a^9 z^3-5 a^9 z+2 a^9 z^{-1} +a^8 z^8-4 a^8 z^6+2 a^8 z^4+4 a^8 z^2-3 a^8+a^7 z^7-6 a^7 z^5+11 a^7 z^3-7 a^7 z+2 a^7 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 \
-9-8-7-6-5-4-3-2-10χ
-6         11
-8        110
-10       2  2
-12     121  0
-14     42   2
-16   122    1
-18   33     0
-20  22      0
-22 13       -2
-24 1        1
-261         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-8 i=-6 i=-4
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4} {\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 {\mathbb Z}
r=0 {\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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L11n9

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L11n11