L10a147

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L10a146.gif

L10a146

L10a148.gif

L10a148

Contents

L10a147.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a147's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X16,5,17,6 X20,10,13,9 X18,8,19,7 X8,14,9,13 X12,15,5,16 X10,20,11,19 X2,11,3,12 X4,18,1,17
Gauss code {1, -9, 2, -10}, {3, -1, 5, -6, 4, -8, 9, -7}, {6, -2, 7, -3, 10, -5, 8, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a147 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(3)^2 t(2)^3-t(3)^2 t(2)^3-t(1) t(3) t(2)^3+t(3) t(2)^3+t(1) t(3)^3 t(2)^2-t(3)^3 t(2)^2-3 t(1) t(3)^2 t(2)^2+3 t(3)^2 t(2)^2-t(1) t(2)^2+4 t(1) t(3) t(2)^2-3 t(3) t(2)^2+t(2)^2-t(1) t(3)^3 t(2)+t(3)^3 t(2)+3 t(1) t(3)^2 t(2)-4 t(3)^2 t(2)+t(1) t(2)-3 t(1) t(3) t(2)+3 t(3) t(2)-t(2)-t(1) t(3)^2+t(3)^2+t(1) t(3)-t(3)}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}} (db)
Jones polynomial q^6-3 q^5+6 q^4+ q^{-4} -9 q^3-4 q^{-3} +14 q^2+8 q^{-2} -13 q-11 q^{-1} +14 (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^6 a^{-2} -z^6+a^2 z^4-3 z^4 a^{-2} +z^4 a^{-4} -2 z^4+a^2 z^2-4 z^2 a^{-2} +2 z^2 a^{-4} -3 a^{-2} + a^{-4} +2-2 a^{-2} z^{-2} + a^{-4} z^{-2} + z^{-2} (db)
Kauffman polynomial 2 z^9 a^{-1} +2 z^9 a^{-3} +10 z^8 a^{-2} +4 z^8 a^{-4} +6 z^8+9 a z^7+8 z^7 a^{-1} +2 z^7 a^{-3} +3 z^7 a^{-5} +8 a^2 z^6-23 z^6 a^{-2} -11 z^6 a^{-4} +z^6 a^{-6} -3 z^6+4 a^3 z^5-11 a z^5-19 z^5 a^{-1} -13 z^5 a^{-3} -9 z^5 a^{-5} +a^4 z^4-9 a^2 z^4+20 z^4 a^{-2} +11 z^4 a^{-4} -3 z^4 a^{-6} -4 z^4-2 a^3 z^3+3 a z^3+8 z^3 a^{-1} +10 z^3 a^{-3} +7 z^3 a^{-5} +3 a^2 z^2-13 z^2 a^{-2} -7 z^2 a^{-4} +2 z^2 a^{-6} -z^2-3 z a^{-1} -3 z a^{-3} +5 a^{-2} +3 a^{-4} +3+2 a^{-1} z^{-1} +2 a^{-3} z^{-1} -2 a^{-2} z^{-2} - a^{-4} z^{-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 \
-4-3-2-10123456χ
13          11
11         2 -2
9        41 3
7       63  -3
5      83   5
3     56    1
1    98     1
-1   58      3
-3  36       -3
-5 15        4
-7 3         -3
-91          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=6 {\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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See/edit the Link Page master template (intermediate).

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L10a146

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