L10a72

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L10a71.gif

L10a71

L10a73.gif

L10a73

Contents

L10a72.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a72 at Knotilus!


Link Presentations

[edit Notes on L10a72's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X14,17,15,18 X16,5,17,6 X4,15,5,16 X18,11,19,12 X20,13,7,14 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, -6, 8, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a72 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(1) t(2)^4-t(2)^4+2 t(1)^2 t(2)^3-4 t(1) t(2)^3+2 t(2)^3-2 t(1)^2 t(2)^2+5 t(1) t(2)^2-2 t(2)^2+2 t(1)^2 t(2)-4 t(1) t(2)+2 t(2)-t(1)^2+2 t(1)}{t(1) t(2)^2} (db)
Jones polynomial -\frac{5}{q^{9/2}}+\frac{2}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{3}{q^{23/2}}-\frac{5}{q^{21/2}}+\frac{8}{q^{19/2}}-\frac{10}{q^{17/2}}+\frac{10}{q^{15/2}}-\frac{10}{q^{13/2}}+\frac{7}{q^{11/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^{11} z^3+a^{11} z-a^{11} z^{-1} -a^9 z^5-a^9 z^3+3 a^9 z+3 a^9 z^{-1} -2 a^7 z^5-6 a^7 z^3-5 a^7 z-2 a^7 z^{-1} -a^5 z^5-3 a^5 z^3-2 a^5 z (db)
Kauffman polynomial -z^5 a^{15}+2 z^3 a^{15}-3 z^6 a^{14}+7 z^4 a^{14}-3 z^2 a^{14}-4 z^7 a^{13}+9 z^5 a^{13}-6 z^3 a^{13}+2 z a^{13}-3 z^8 a^{12}+4 z^6 a^{12}-2 z^2 a^{12}+a^{12}-z^9 a^{11}-4 z^7 a^{11}+12 z^5 a^{11}-13 z^3 a^{11}+5 z a^{11}-a^{11} z^{-1} -5 z^8 a^{10}+9 z^6 a^{10}-5 z^4 a^{10}-5 z^2 a^{10}+3 a^{10}-z^9 a^9-3 z^7 a^9+10 z^5 a^9-16 z^3 a^9+12 z a^9-3 a^9 z^{-1} -2 z^8 a^8+6 z^4 a^8-7 z^2 a^8+3 a^8-3 z^7 a^7+7 z^5 a^7-8 z^3 a^7+7 z a^7-2 a^7 z^{-1} -2 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-2 z a^5 (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
-4          11
-6         21-1
-8        3  3
-10       42  -2
-12      63   3
-14     44    0
-16    66     0
-18   35      2
-20  25       -3
-22 13        2
-24 2         -2
-261          1
Integral Khovanov Homology

(db, data source)

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