L11n55

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L11n54.gif

L11n54

L11n56.gif

L11n56

Contents

L11n55.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n55's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,11,17,12 X7,15,8,14 X15,9,16,8 X20,13,21,14 X22,17,5,18 X18,21,19,22 X12,19,13,20 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, -4, 5, 11, -2, 3, -9, 6, 4, -5, -3, 7, -8, 9, -6, 8, -7}
A Braid Representative
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A Morse Link Presentation L11n55 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 t(1) t(2)^3-3 t(2)^3-6 t(1) t(2)^2+6 t(2)^2+6 t(1) t(2)-6 t(2)-3 t(1)+2}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial \frac{11}{q^{9/2}}-\frac{12}{q^{7/2}}+\frac{9}{q^{5/2}}-\frac{6}{q^{3/2}}+\frac{2}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{11}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^9 z^{-1} -a^7 z^3+2 a^7 z+3 a^7 z^{-1} +a^5 z^5-3 a^5 z-2 a^5 z^{-1} +a^3 z^5+a^3 z^3-a z^3-a z (db)
Kauffman polynomial -3 z^4 a^{10}+4 z^2 a^{10}-a^{10}-z^7 a^9-3 z^5 a^9+6 z^3 a^9-3 z a^9+a^9 z^{-1} -2 z^8 a^8+z^6 a^8-4 z^4 a^8+8 z^2 a^8-3 a^8-z^9 a^7-3 z^7 a^7+z^5 a^7+7 z^3 a^7-8 z a^7+3 a^7 z^{-1} -5 z^8 a^6+4 z^6 a^6+3 z^2 a^6-3 a^6-z^9 a^5-6 z^7 a^5+11 z^5 a^5-3 z^3 a^5-3 z a^5+2 a^5 z^{-1} -3 z^8 a^4+7 z^4 a^4-4 z^2 a^4-4 z^7 a^3+6 z^5 a^3-2 z^3 a^3+z a^3-3 z^6 a^2+6 z^4 a^2-3 z^2 a^2-z^5 a+2 z^3 a-z a (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         11
0        2 -2
-2       41 3
-4      63  -3
-6     63   3
-8    56    1
-10   66     0
-12  36      3
-14 25       -3
-16 3        3
-182         -2
Integral Khovanov Homology

(db, data source)

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