L10a42

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L10a41.gif

L10a41

L10a43.gif

L10a43

Contents

L10a42.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a42 at Knotilus!


Link Presentations

[edit Notes on L10a42's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X18,14,19,13 X14,18,15,17 X4,19,1,20 X10,5,11,6 X12,3,13,4 X20,11,5,12 X2,9,3,10 X8,15,9,16
Gauss code {1, -9, 7, -5}, {6, -1, 2, -10, 9, -6, 8, -7, 3, -4, 10, -2, 4, -3, 5, -8}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
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BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a42 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u v^3-9 u v^2+8 u v-2 u-2 v^3+8 v^2-9 v+2}{\sqrt{u} v^{3/2}} (db)
Jones polynomial -\sqrt{q}+\frac{4}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{14}{q^{7/2}}+\frac{14}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{10}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z-2 a^7 z^3-a^7 z+a^5 z^5+a^5 z^{-1} +a^3 z^5-2 a^3 z-a^3 z^{-1} -a z^3 (db)
Kauffman polynomial a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-6 a^{10} z^4+3 a^{10} z^2+4 a^9 z^7-5 a^9 z^5+4 a^8 z^8-4 a^8 z^6+a^8 z^4-a^8 z^2+2 a^7 z^9+4 a^7 z^7-11 a^7 z^5+8 a^7 z^3-a^7 z+10 a^6 z^8-21 a^6 z^6+19 a^6 z^4-6 a^6 z^2+2 a^5 z^9+7 a^5 z^7-20 a^5 z^5+14 a^5 z^3-2 a^5 z+a^5 z^{-1} +6 a^4 z^8-10 a^4 z^6+6 a^4 z^4-2 a^4 z^2-a^4+7 a^3 z^7-14 a^3 z^5+7 a^3 z^3-2 a^3 z+a^3 z^{-1} +4 a^2 z^6-6 a^2 z^4+a z^5-a z^3 (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 \
-8-7-6-5-4-3-2-1012χ
2          11
0         3 -3
-2        51 4
-4       64  -2
-6      84   4
-8     66    0
-10    78     -1
-12   47      3
-14  26       -4
-16 14        3
-18 2         -2
-201          1
Integral Khovanov Homology

(db, data source)

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

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L10a41

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L10a43