L11n90

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L11n89.gif

L11n89

L11n91.gif

L11n91

Contents

L11n90.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n90 at Knotilus!


Link Presentations

[edit Notes on L11n90's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X7,16,8,17 X17,22,18,5 X14,9,15,10 X10,20,11,19 X21,9,22,8 X18,14,19,13 X20,15,21,16 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, -3, 7, 5, -6, 11, -2, 8, -5, 9, 3, -4, -8, 6, -9, -7, 4}
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n90 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(2)-2) (t(1)+t(2)) (2 t(2)-1)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -\frac{3}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{5}{q^{5/2}}+\frac{6}{q^{7/2}}-\frac{6}{q^{9/2}}+\frac{5}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^9 z^{-1} -3 z a^7-2 a^7 z^{-1} +2 z^3 a^5+2 z a^5+a^5 z^{-1} +2 z^3 a^3+2 z a^3+a^3 z^{-1} -3 z a-a z^{-1} (db)
Kauffman polynomial -z^7 a^9+5 z^5 a^9-8 z^3 a^9+5 z a^9-a^9 z^{-1} -2 z^8 a^8+9 z^6 a^8-11 z^4 a^8+4 z^2 a^8-a^8-z^9 a^7+15 z^5 a^7-25 z^3 a^7+13 z a^7-2 a^7 z^{-1} -5 z^8 a^6+21 z^6 a^6-25 z^4 a^6+13 z^2 a^6-3 a^6-z^9 a^5-z^7 a^5+15 z^5 a^5-19 z^3 a^5+9 z a^5-a^5 z^{-1} -3 z^8 a^4+12 z^6 a^4-17 z^4 a^4+12 z^2 a^4-2 a^4-2 z^7 a^3+5 z^5 a^3-2 z^3 a^3-3 z a^3+a^3 z^{-1} -3 z^4 a^2+3 z^2 a^2-a^2-4 z a+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 \
-8-7-6-5-4-3-2-10χ
0        33
-2       21-1
-4      32 1
-6     32  -1
-8    33   0
-10   34    1
-12  12     -1
-14 13      2
-16 1       -1
-181        1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}

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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L11n89

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L11n91