L10a149

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L10a148.gif

L10a148

L10a150.gif

L10a150

Contents

L10a149.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a149's Link Presentations]

Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X16,6,17,5 X8493 X20,18,15,17 X10,20,11,19 X18,10,19,9 X14,16,5,15 X2,12,3,11
Gauss code {1, -10, 5, -3}, {9, -4, 6, -8, 7, -6}, {4, -1, 2, -5, 8, -7, 10, -2, 3, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a149 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(3)-1)^2 (2 t(3) t(2)-t(2)-t(3)+2)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial -q^8+5 q^7-9 q^6+13 q^5-15 q^4+17 q^3-14 q^2+12 q-6+3 q^{-1} - q^{-2} (db)
Signature 2 (db)
HOMFLY-PT polynomial -z^4 a^{-6} + a^{-6} z^{-2} +2 a^{-6} +z^6 a^{-4} +z^4 a^{-4} -4 z^2 a^{-4} -2 a^{-4} z^{-2} -7 a^{-4} +z^6 a^{-2} +3 z^4 a^{-2} +6 z^2 a^{-2} + a^{-2} z^{-2} +6 a^{-2} -z^4-2 z^2-1 (db)
Kauffman polynomial z^5 a^{-9} +5 z^6 a^{-8} -6 z^4 a^{-8} + a^{-8} +9 z^7 a^{-7} -15 z^5 a^{-7} +4 z^3 a^{-7} +7 z^8 a^{-6} -5 z^6 a^{-6} -8 z^4 a^{-6} +5 z^2 a^{-6} + a^{-6} z^{-2} -3 a^{-6} +2 z^9 a^{-5} +13 z^7 a^{-5} -28 z^5 a^{-5} +9 z^3 a^{-5} +5 z a^{-5} -2 a^{-5} z^{-1} +11 z^8 a^{-4} -12 z^6 a^{-4} -9 z^4 a^{-4} +18 z^2 a^{-4} +2 a^{-4} z^{-2} -10 a^{-4} +2 z^9 a^{-3} +8 z^7 a^{-3} -16 z^5 a^{-3} +4 z^3 a^{-3} +7 z a^{-3} -2 a^{-3} z^{-1} +4 z^8 a^{-2} +z^6 a^{-2} -13 z^4 a^{-2} +18 z^2 a^{-2} + a^{-2} z^{-2} -9 a^{-2} +4 z^7 a^{-1} +a z^5-3 z^5 a^{-1} -2 a z^3-3 z^3 a^{-1} +a z+3 z a^{-1} +3 z^6-6 z^4+5 z^2-2 (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 \
-3-2-101234567χ
17          1-1
15         4 4
13        51 -4
11       84  4
9      97   -2
7     86    2
5    69     3
3   68      -2
1  28       6
-1 14        -3
-3 2         2
-51          -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\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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L10a148

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