L11n60

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L11n59.gif

L11n59

L11n61.gif

L11n61

Contents

L11n60.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n60 at Knotilus!


Link Presentations

[edit Notes on L11n60's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,8,15,7 X11,19,12,18 X22,19,5,20 X20,15,21,16 X16,21,17,22 X17,13,18,12 X8,14,9,13 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, -4, 8, 9, -3, 6, -7, -8, 4, 5, -6, 7, -5}
A Braid Representative
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A Morse Link Presentation L11n60 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^5+t(1) t(2)^4-3 t(2)^4-4 t(1) t(2)^3+6 t(2)^3+6 t(1) t(2)^2-4 t(2)^2-3 t(1) t(2)+t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -q^{3/2}+4 \sqrt{q}-\frac{7}{\sqrt{q}}+\frac{9}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{9}{q^{7/2}}-\frac{9}{q^{9/2}}+\frac{6}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7-a^7 z^{-1} +3 z^3 a^5+6 z a^5+4 a^5 z^{-1} -2 z^5 a^3-7 z^3 a^3-10 z a^3-4 a^3 z^{-1} +3 z^3 a+4 z a+a z^{-1} -z a^{-1} (db)
Kauffman polynomial a^8 z^6-3 a^8 z^4+3 a^8 z^2-a^8+3 a^7 z^7-9 a^7 z^5+8 a^7 z^3-4 a^7 z+a^7 z^{-1} +3 a^6 z^8-4 a^6 z^6-9 a^6 z^4+11 a^6 z^2-4 a^6+a^5 z^9+7 a^5 z^7-29 a^5 z^5+29 a^5 z^3-15 a^5 z+4 a^5 z^{-1} +6 a^4 z^8-11 a^4 z^6-4 a^4 z^4+14 a^4 z^2-7 a^4+a^3 z^9+6 a^3 z^7-23 a^3 z^5+28 a^3 z^3-14 a^3 z+4 a^3 z^{-1} +3 a^2 z^8-6 a^2 z^6+6 a^2 z^4+4 a^2 z^2-4 a^2+2 a z^7-3 a z^5+8 a z^3+z^3 a^{-1} -4 a z-z a^{-1} +a z^{-1} +4 z^4-2 z^2-1 (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         11
2        3 -3
0       41 3
-2      64  -2
-4     53   2
-6    46    2
-8   55     0
-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}^{5}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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