L10a146

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L10a145.gif

L10a145

L10a147.gif

L10a147

Contents

L10a146.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a146's Link Presentations]

Planar diagram presentation X6172 X14,4,15,3 X20,12,13,11 X18,8,19,7 X16,10,17,9 X8,18,9,17 X10,14,11,13 X12,20,5,19 X2536 X4,16,1,15
Gauss code {1, -9, 2, -10}, {9, -1, 4, -6, 5, -7, 3, -8}, {7, -2, 10, -5, 6, -4, 8, -3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gif
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A Morse Link Presentation L10a146 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u v^3 w+2 u v^2 w^2-3 u v^2 w+u v^2+u v w^3-3 u v w^2+2 u v w-u w^3+u w^2-v^3 w+v^3-2 v^2 w^2+3 v^2 w-v^2-v w^3+3 v w^2-2 v w-w^2}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial q^9-2 q^8+5 q^7-7 q^6+10 q^5-9 q^4+10 q^3-8 q^2+5 q+ q^{-1} -2 (db)
Signature 2 (db)
HOMFLY-PT polynomial -z^4 a^{-2} -2 z^4 a^{-4} -z^4 a^{-6} -2 z^2 a^{-4} -z^2 a^{-6} +z^2 a^{-8} +z^2+ a^{-4} -3 a^{-6} + a^{-8} +1+ a^{-4} z^{-2} -2 a^{-6} z^{-2} + a^{-8} z^{-2} (db)
Kauffman polynomial z^6 a^{-10} -4 z^4 a^{-10} +4 z^2 a^{-10} +2 z^7 a^{-9} -6 z^5 a^{-9} +4 z^3 a^{-9} +3 z^8 a^{-8} -11 z^6 a^{-8} +18 z^4 a^{-8} -19 z^2 a^{-8} - a^{-8} z^{-2} +8 a^{-8} +z^9 a^{-7} +3 z^7 a^{-7} -15 z^5 a^{-7} +17 z^3 a^{-7} -11 z a^{-7} +2 a^{-7} z^{-1} +6 z^8 a^{-6} -18 z^6 a^{-6} +28 z^4 a^{-6} -30 z^2 a^{-6} -2 a^{-6} z^{-2} +13 a^{-6} +z^9 a^{-5} +5 z^7 a^{-5} -17 z^5 a^{-5} +21 z^3 a^{-5} -11 z a^{-5} +2 a^{-5} z^{-1} +3 z^8 a^{-4} -3 z^6 a^{-4} +3 z^4 a^{-4} -5 z^2 a^{-4} - a^{-4} z^{-2} +5 a^{-4} +4 z^7 a^{-3} -6 z^5 a^{-3} +6 z^3 a^{-3} +3 z^6 a^{-2} -2 z^4 a^{-2} +2 z^5 a^{-1} -2 z^3 a^{-1} +z^4-2 z^2+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 \
-2-1012345678χ
19          11
17         21-1
15        3  3
13       42  -2
11      63   3
9     45    1
7    65     1
5   24      2
3  36       -3
1 14        3
-1 1         -1
-31          1
Integral Khovanov Homology

(db, data source)

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