L11n272

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L11n271.gif

L11n271

L11n273.gif

L11n273

Contents

L11n272.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n272's Link Presentations]

Planar diagram presentation X6172 X3,11,4,10 X7,17,8,16 X15,5,16,8 X18,11,19,12 X22,17,9,18 X12,21,13,22 X20,13,21,14 X14,19,15,20 X2536 X9,1,10,4
Gauss code {1, -10, -2, 11}, {10, -1, -3, 4}, {-11, 2, 5, -7, 8, -9, -4, 3, 6, -5, 9, -8, 7, -6}
A Braid Representative
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A Morse Link Presentation L11n272 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(3)^3-2 t(1) t(3)^2-2 t(2) t(3)^2-t(3)^2+2 t(1) t(3)+t(1) t(2) t(3)+2 t(2) t(3)-2 t(1) t(2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial q^3-q^2+2 q+1+ q^{-1} + q^{-2} - q^{-3} +2 q^{-4} -2 q^{-5} + q^{-6} - q^{-7} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 \left(-z^2\right)-a^6 z^{-2} -2 a^6+a^4 z^4+4 a^4 z^2+3 a^4 z^{-2} +6 a^4-a^2 z^2-2 a^2 z^{-2} +z^2 a^{-2} + a^{-2} z^{-2} -3 a^2+2 a^{-2} -z^4-4 z^2- z^{-2} -3 (db)
Kauffman polynomial a^7 z^7-6 a^7 z^5+11 a^7 z^3-7 a^7 z+2 a^7 z^{-1} +a^6 z^8-5 a^6 z^6+6 a^6 z^4-2 a^6 z^2-a^6 z^{-2} +a^6+3 a^5 z^7-18 a^5 z^5+33 a^5 z^3-27 a^5 z+8 a^5 z^{-1} +a^4 z^8-5 a^4 z^6+5 a^4 z^4-4 a^4 z^2-3 a^4 z^{-2} +5 a^4+3 a^3 z^7-21 a^3 z^5+42 a^3 z^3-34 a^3 z+10 a^3 z^{-1} +a^2 z^8-6 a^2 z^6+z^6 a^{-2} +7 a^2 z^4-5 z^4 a^{-2} -3 a^2 z^2+6 z^2 a^{-2} -2 a^2 z^{-2} + a^{-2} z^{-2} +4 a^2-4 a^{-2} +2 a z^7+z^7 a^{-1} -13 a z^5-4 z^5 a^{-1} +20 a z^3-10 a z+4 z a^{-1} +2 a z^{-1} -2 a^{-1} z^{-1} +z^8-5 z^6+3 z^4+5 z^2+ z^{-2} -3 (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 \
-7-6-5-4-3-2-101234χ
7           11
5          110
3         1  1
1       311  3
-1      251   2
-3     1 1    2
-5    132     0
-7   21       1
-9   11       0
-11 12         -1
-13            0
-151           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1 i=1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}_2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{2} {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{5} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}_2 {\mathbb Z}
r=4 {\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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L11n271

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L11n273