L10a79

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L10a78.gif

L10a78

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L10a80

Contents

L10a79.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a79's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X20,13,7,14 X14,9,15,10 X10,19,11,20 X16,6,17,5 X18,16,19,15 X2738 X4,11,5,12 X6,18,1,17
Gauss code {1, -8, 2, -9, 6, -10}, {8, -1, 4, -5, 9, -2, 3, -4, 7, -6, 10, -7, 5, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L10a79 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^3-5 u^2 v^2+4 u^2 v-u^2+u v^4-5 u v^3+9 u v^2-5 u v+u-v^4+4 v^3-5 v^2+2 v}{u v^2} (db)
Jones polynomial q^{5/2}-4 q^{3/2}+8 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{14}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{11}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7-a^7 z^{-1} +3 z^3 a^5+5 z a^5+3 a^5 z^{-1} -2 z^5 a^3-5 z^3 a^3-6 z a^3-2 a^3 z^{-1} -z^5 a+z a+z^3 a^{-1} (db)
Kauffman polynomial a^8 z^6-3 a^8 z^4+3 a^8 z^2-a^8+3 a^7 z^7-8 a^7 z^5+7 a^7 z^3-3 a^7 z+a^7 z^{-1} +4 a^6 z^8-7 a^6 z^6-2 a^6 z^4+7 a^6 z^2-3 a^6+2 a^5 z^9+6 a^5 z^7-26 a^5 z^5+24 a^5 z^3-10 a^5 z+3 a^5 z^{-1} +11 a^4 z^8-20 a^4 z^6+3 a^4 z^4+6 a^4 z^2-3 a^4+2 a^3 z^9+13 a^3 z^7-36 a^3 z^5+27 a^3 z^3-9 a^3 z+2 a^3 z^{-1} +7 a^2 z^8-4 a^2 z^6-7 a^2 z^4+z^4 a^{-2} +4 a^2 z^2+10 a z^7-14 a z^5+4 z^5 a^{-1} +8 a z^3-2 z^3 a^{-1} -2 a z+8 z^6-8 z^4+2 z^2 (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 \
-7-6-5-4-3-2-10123χ
6          1-1
4         3 3
2        51 -4
0       73  4
-2      86   -2
-4     86    2
-6    69     3
-8   57      -2
-10  26       4
-12 15        -4
-14 2         2
-161          -1
Integral Khovanov Homology

(db, data source)

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

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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