L10a73

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L10a72.gif

L10a72

L10a74.gif

L10a74

Contents

L10a73.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a73 at Knotilus!


Link Presentations

[edit Notes on L10a73's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X20,11,7,12 X18,13,19,14 X12,19,13,20 X2738 X6,9,1,10
Gauss code {1, -9, 2, -5, 4, -10}, {9, -1, 10, -2, 6, -8, 7, -3, 5, -4, 3, -7, 8, -6}
A Braid Representative
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A Morse Link Presentation L10a73 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{3 t(2) t(1)^2-2 t(1)^2+3 t(2)^2 t(1)-5 t(2) t(1)+3 t(1)-2 t(2)^2+3 t(2)}{t(1) t(2)} (db)
Jones polynomial \frac{5}{q^{9/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{23/2}}+\frac{2}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{5}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{7}{q^{11/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^{11} z-a^9 z^3-2 a^7 z^3-a^7 z+a^7 z^{-1} -2 a^5 z^3-2 a^5 z-a^5 z^{-1} -a^3 z^3-a^3 z (db)
Kauffman polynomial a^{13} z^7-5 a^{13} z^5+7 a^{13} z^3-2 a^{13} z+2 a^{12} z^8-10 a^{12} z^6+15 a^{12} z^4-7 a^{12} z^2+a^{11} z^9-2 a^{11} z^7-4 a^{11} z^5+7 a^{11} z^3-a^{11} z+4 a^{10} z^8-15 a^{10} z^6+15 a^{10} z^4-5 a^{10} z^2+a^9 z^9-6 a^9 z^5+3 a^9 z^3+2 a^8 z^8-2 a^8 z^6-4 a^8 z^4+2 a^8 z^2+3 a^7 z^7-4 a^7 z^5-a^7 z^3+3 a^7 z-a^7 z^{-1} +3 a^6 z^6-2 a^6 z^4-a^6 z^2+a^6+3 a^5 z^5-3 a^5 z^3+3 a^5 z-a^5 z^{-1} +2 a^4 z^4-a^4 z^2+a^3 z^3-a^3 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 \
-10-9-8-7-6-5-4-3-2-10χ
-2          11
-4         21-1
-6        2  2
-8       32  -1
-10      42   2
-12     23    1
-14    44     0
-16   23      1
-18  13       -2
-20 12        1
-22 1         -1
-241          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
r=-10 {\mathbb Z}
r=-9 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L10a72

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L10a74