L10a102

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L10a101.gif

L10a101

L10a103.gif

L10a103

Contents

L10a102.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a102 at Knotilus!


Link Presentations

[edit Notes on L10a102's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X16,5,17,6 X20,13,9,14 X14,19,15,20 X18,7,19,8 X8,9,1,10 X4,15,5,16 X6,17,7,18
Gauss code {1, -2, 3, -9, 4, -10, 7, -8}, {8, -1, 2, -3, 5, -6, 9, -4, 10, -7, 6, -5}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a102 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1) t(2)+1) \left(2 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+2 t(2) t(1)-t(1)-t(2)+2\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{6}{q^{19/2}}+\frac{6}{q^{21/2}}-\frac{4}{q^{23/2}}+\frac{2}{q^{25/2}}-\frac{1}{q^{27/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial z^5 a^{11}+4 z^3 a^{11}+4 z a^{11}-z^7 a^9-5 z^5 a^9-7 z^3 a^9-2 z a^9+a^9 z^{-1} -z^7 a^7-6 z^5 a^7-11 z^3 a^7-6 z a^7-a^7 z^{-1} (db)
Kauffman polynomial a^{17} z^3-a^{17} z+2 a^{16} z^4-a^{16} z^2+3 a^{15} z^5-2 a^{15} z^3+a^{15} z+4 a^{14} z^6-7 a^{14} z^4+6 a^{14} z^2+3 a^{13} z^7-4 a^{13} z^5+a^{13} z+2 a^{12} z^8-3 a^{12} z^6-a^{12} z^2+a^{11} z^9-2 a^{11} z^7+4 a^{11} z^5-11 a^{11} z^3+5 a^{11} z+3 a^{10} z^8-11 a^{10} z^6+12 a^{10} z^4-6 a^{10} z^2+a^9 z^9-4 a^9 z^7+5 a^9 z^5-3 a^9 z^3+a^9 z^{-1} +a^8 z^8-4 a^8 z^6+3 a^8 z^4+2 a^8 z^2-a^8+a^7 z^7-6 a^7 z^5+11 a^7 z^3-6 a^7 z+a^7 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 \
-10-9-8-7-6-5-4-3-2-10χ
-6          11
-8         110
-10        2  2
-12       21  -1
-14      42   2
-16     22    0
-18    44     0
-20   22      0
-22  24       -2
-24 13        2
-26 1         -1
-281          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-8 i=-6
r=-10 {\mathbb Z}
r=-9 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L10a101.gif

L10a101

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L10a103