L11n242

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L11n241.gif

L11n241

L11n243.gif

L11n243

Contents

L11n242.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n242 at Knotilus!


Link Presentations

[edit Notes on L11n242's Link Presentations]

Planar diagram presentation X12,1,13,2 X16,8,17,7 X10,5,1,6 X6374 X4,9,5,10 X13,18,14,19 X19,22,20,11 X15,21,16,20 X21,15,22,14 X2,11,3,12 X8,18,9,17
Gauss code {1, -10, 4, -5, 3, -4, 2, -11, 5, -3}, {10, -1, -6, 9, -8, -2, 11, 6, -7, 8, -9, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
A Morse Link Presentation L11n242 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(t(2) t(1)^2+t(2)^2 t(1)-4 t(2) t(1)+t(1)+t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{9}{q^{9/2}}+\frac{10}{q^{7/2}}-\frac{11}{q^{5/2}}-2 q^{3/2}+\frac{10}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{6}{q^{11/2}}+4 \sqrt{q}-\frac{8}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)-a^7 z^{-1} +3 a^5 z^3+6 a^5 z+4 a^5 z^{-1} -2 a^3 z^5-7 a^3 z^3-11 a^3 z-6 a^3 z^{-1} +4 a z^3+8 a z+5 a z^{-1} -2 z a^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -a^5 z^9-a^3 z^9-3 a^6 z^8-7 a^4 z^8-4 a^2 z^8-3 a^7 z^7-8 a^5 z^7-9 a^3 z^7-4 a z^7-a^8 z^6+4 a^6 z^6+15 a^4 z^6+9 a^2 z^6-z^6+9 a^7 z^5+33 a^5 z^5+36 a^3 z^5+12 a z^5+3 a^8 z^4+8 a^6 z^4-3 a^4 z^4-10 a^2 z^4-2 z^4-8 a^7 z^3-35 a^5 z^3-50 a^3 z^3-26 a z^3-3 z^3 a^{-1} -3 a^8 z^2-9 a^6 z^2-7 a^4 z^2+z^2+4 a^7 z+18 a^5 z+29 a^3 z+20 a z+5 z a^{-1} +a^8+3 a^6+3 a^4+a^2+1-a^7 z^{-1} -4 a^5 z^{-1} -6 a^3 z^{-1} -5 a z^{-1} -2 a^{-1} 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
4         22
2        2 -2
0       62 4
-2      64  -2
-4     54   1
-6    56    1
-8   45     -1
-10  25      3
-12 14       -3
-14 2        2
-161         -1
Integral Khovanov Homology

(db, data source)

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

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L11n241.gif

L11n241

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L11n243