L11n71

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L11n70.gif

L11n70

L11n72.gif

L11n72

Contents

L11n71.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n71 at Knotilus!


Link Presentations

[edit Notes on L11n71's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X7,16,8,17 X17,22,18,5 X11,18,12,19 X21,12,22,13 X13,20,14,21 X19,14,20,15 X15,8,16,9 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, -3, 9, 11, -2, -5, 6, -7, 8, -9, 3, -4, 5, -8, 7, -6, 4}
A Braid Representative
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A Morse Link Presentation L11n71 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u v^3-u v^2-2 u v+2 u+2 v^3-2 v^2-v+2}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -\frac{1}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{5/2}}+\frac{1}{q^{25/2}}-\frac{1}{q^{23/2}}+\frac{2}{q^{21/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{17/2}}-\frac{1}{q^{15/2}}-\frac{1}{q^{13/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{13} z^{-1} +2 z a^{11}+a^{11} z^{-1} +3 z a^9+2 a^9 z^{-1} -z^5 a^7-5 z^3 a^7-5 z a^7-2 a^7 z^{-1} -z^5 a^5-4 z^3 a^5-2 z a^5 (db)
Kauffman polynomial a^{14} z^8-7 a^{14} z^6+16 a^{14} z^4-14 a^{14} z^2+4 a^{14}+a^{13} z^9-6 a^{13} z^7+10 a^{13} z^5-5 a^{13} z^3+a^{13} z-a^{13} z^{-1} +3 a^{12} z^8-20 a^{12} z^6+41 a^{12} z^4-33 a^{12} z^2+9 a^{12}+a^{11} z^9-5 a^{11} z^7+3 a^{11} z^5+5 a^{11} z^3-2 a^{11} z-a^{11} z^{-1} +2 a^{10} z^8-13 a^{10} z^6+22 a^{10} z^4-13 a^{10} z^2+4 a^{10}+2 a^9 z^7-14 a^9 z^5+24 a^9 z^3-13 a^9 z+2 a^9 z^{-1} +a^8 z^6-7 a^8 z^4+8 a^8 z^2-2 a^8+a^7 z^7-6 a^7 z^5+10 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +a^6 z^6-4 a^6 z^4+2 a^6 z^2+a^5 z^5-4 a^5 z^3+2 a^5 z (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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-4           11
-6          110
-8        11  0
-10       111  1
-12      241   1
-14     1 1    2
-16    132     0
-18   21       1
-20   11       0
-22 12         -1
-24            0
-261           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4 i=-2
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{2}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-7 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}_2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2} {\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4} {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\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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L11n72