L10a41

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L10a40.gif

L10a40

L10a42.gif

L10a42

Contents

L10a41.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a41 at Knotilus!


Link Presentations

[edit Notes on L10a41's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X16,10,17,9 X14,12,15,11 X10,16,11,15 X20,17,5,18 X18,7,19,8 X8,19,9,20 X2536 X4,14,1,13
Gauss code {1, -9, 2, -10}, {9, -1, 7, -8, 3, -5, 4, -2, 10, -4, 5, -3, 6, -7, 8, -6}
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.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gif
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A Morse Link Presentation L10a41 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-3 t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial q^{9/2}-\frac{2}{q^{9/2}}-3 q^{7/2}+\frac{4}{q^{7/2}}+5 q^{5/2}-\frac{7}{q^{5/2}}-7 q^{3/2}+\frac{8}{q^{3/2}}+\frac{1}{q^{11/2}}+8 \sqrt{q}-\frac{10}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^5-a^5 z^{-1} +2 z^3 a^3+4 z a^3+3 a^3 z^{-1} -z^5 a-2 z^3 a-3 z a-2 a z^{-1} -z^5 a^{-1} -2 z^3 a^{-1} -z a^{-1} +z^3 a^{-3} +z a^{-3} (db)
Kauffman polynomial a^6 z^4-2 a^6 z^2+a^6+2 a^5 z^5-3 a^5 z^3+2 a^5 z-a^5 z^{-1} +2 a^4 z^6+z^6 a^{-4} +a^4 z^4-3 z^4 a^{-4} -5 a^4 z^2+2 z^2 a^{-4} +3 a^4+2 a^3 z^7+3 z^7 a^{-3} +2 a^3 z^5-10 z^5 a^{-3} -7 a^3 z^3+8 z^3 a^{-3} +7 a^3 z-2 z a^{-3} -3 a^3 z^{-1} +2 a^2 z^8+3 z^8 a^{-2} -a^2 z^6-8 z^6 a^{-2} +2 a^2 z^4+3 z^4 a^{-2} -4 a^2 z^2+z^2 a^{-2} +3 a^2+a z^9+z^9 a^{-1} +2 a z^7+3 z^7 a^{-1} -5 a z^5-15 z^5 a^{-1} +12 z^3 a^{-1} +5 a z-2 z a^{-1} -2 a z^{-1} +5 z^8-12 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 \
-5-4-3-2-1012345χ
10          1-1
8         2 2
6        31 -2
4       42  2
2      43   -1
0     64    2
-2    46     2
-4   34      -1
-6  14       3
-8 13        -2
-10 1         1
-121          -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=5 {\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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L10a40

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L10a42