L11n225

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L11n224.gif

L11n224

L11n226.gif

L11n226

Contents

L11n225.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n225 at Knotilus!


Link Presentations

[edit Notes on L11n225's Link Presentations]

Planar diagram presentation X10,1,11,2 X3,12,4,13 X16,9,17,10 X11,20,12,21 X22,15,9,16 X5,14,6,15 X13,4,14,5 X6,20,7,19 X18,8,19,7 X8,18,1,17 X21,3,22,2
Gauss code {1, 11, -2, 7, -6, -8, 9, -10}, {3, -1, -4, 2, -7, 6, 5, -3, 10, -9, 8, 4, -11, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n225 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2) t(1)^2+2 t(2)^2 t(1)-t(2) t(1)+2 t(1)+t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{10}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 \left(-z^3\right)-a^7 z+a^5 z^5+a^5 z^3-2 a^5 z-a^5 z^{-1} +2 a^3 z^5+7 a^3 z^3+8 a^3 z+3 a^3 z^{-1} -2 a z^3-5 a z-2 a z^{-1} (db)
Kauffman polynomial -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+3 z^3 a^9-5 z^6 a^8+7 z^4 a^8-3 z^2 a^8-5 z^7 a^7+8 z^5 a^7-7 z^3 a^7+2 z a^7-3 z^8 a^6+3 z^6 a^6-4 z^4 a^6+4 z^2 a^6-a^6-z^9 a^5-2 z^7 a^5+2 z^5 a^5+4 z^3 a^5-4 z a^5+a^5 z^{-1} -4 z^8 a^4+10 z^6 a^4-14 z^4 a^4+12 z^2 a^4-3 a^4-z^9 a^3+3 z^7 a^3-12 z^5 a^3+24 z^3 a^3-15 z a^3+3 a^3 z^{-1} -z^8 a^2+2 z^6 a^2-2 z^4 a^2+4 z^2 a^2-3 a^2-3 z^5 a+10 z^3 a-9 z a+2 a 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 \
-7-6-5-4-3-2-1012χ
2         22
0        1 -1
-2       52 3
-4      53  -2
-6     53   2
-8    45    1
-10   45     -1
-12  24      2
-14 14       -3
-16 2        2
-181         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{5}
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

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See/edit the Link Page master template (intermediate).

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L11n224.gif

L11n224

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L11n226