L11n409

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L11n408.gif

L11n408

L11n410.gif

L11n410

Contents

L11n409.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n409 at Knotilus!


Link Presentations

[edit Notes on L11n409's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X7,16,8,17 X22,17,19,18 X20,12,21,11 X10,20,11,19 X18,21,5,22 X9,14,10,15 X15,8,16,9 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {6, -5, 7, -4}, {10, -1, -3, 9, -8, -6, 5, -2, 11, 8, -9, 3, 4, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L11n409 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) (t(3)-1) \left(t(3)^2-3 t(3)+1\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial q^4-4 q^3+9 q^2-11 q+14-13 q^{-1} +13 q^{-2} -8 q^{-3} +5 q^{-4} -2 q^{-5} (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^6+3 a^2 z^4+z^4 a^{-2} -3 z^4-2 a^4 z^2+7 a^2 z^2+z^2 a^{-2} -6 z^2-2 a^4+6 a^2+2 a^{-2} -6+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial a^3 z^9+a z^9+a^4 z^8+6 a^2 z^8+5 z^8+3 a^3 z^7+13 a z^7+10 z^7 a^{-1} +2 a^4 z^6-a^2 z^6+9 z^6 a^{-2} +6 z^6+3 a^5 z^5-5 a^3 z^5-25 a z^5-13 z^5 a^{-1} +4 z^5 a^{-3} -9 a^4 z^4-25 a^2 z^4-11 z^4 a^{-2} +z^4 a^{-4} -28 z^4-6 a^5 z^3-5 a^3 z^3+5 a z^3+3 z^3 a^{-1} -z^3 a^{-3} +10 a^4 z^2+30 a^2 z^2+6 z^2 a^{-2} +26 z^2+2 a^5 z+6 a^3 z+6 a z+2 z a^{-1} -4 a^4-12 a^2-4 a^{-2} -11-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2} (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 \
-5-4-3-2-101234χ
9         11
7        3 -3
5       61 5
3      53  -2
1     96   3
-1    89    1
-3   55     0
-5  38      5
-7 25       -3
-9 3        3
-112         -2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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