L10a39

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L10a38.gif

L10a38

L10a40.gif

L10a40

Contents

L10a39.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a39 at Knotilus!


Link Presentations

[edit Notes on L10a39's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,10,15,9 X16,12,17,11 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, -3, 5, -4, 6, -7, 8, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a39 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^2-v+1\right)^2}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -3 q^{9/2}+\frac{1}{q^{9/2}}+7 q^{7/2}-\frac{2}{q^{7/2}}-10 q^{5/2}+\frac{5}{q^{5/2}}+11 q^{3/2}-\frac{9}{q^{3/2}}+q^{11/2}-13 \sqrt{q}+\frac{10}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial z^5 a^{-3} -a^3 z^3+3 z^3 a^{-3} -3 a^3 z+4 z a^{-3} -2 a^3 z^{-1} +2 a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-5 z^5 a^{-1} +8 a z^3-11 z^3 a^{-1} +12 a z-13 z a^{-1} +7 a z^{-1} -7 a^{-1} z^{-1} (db)
Kauffman polynomial -a z^9-z^9 a^{-1} -2 a^2 z^8-4 z^8 a^{-2} -6 z^8-2 a^3 z^7-5 a z^7-10 z^7 a^{-1} -7 z^7 a^{-3} -a^4 z^6+a^2 z^6-2 z^6 a^{-2} -6 z^6 a^{-4} +6 z^6+6 a^3 z^5+19 a z^5+27 z^5 a^{-1} +11 z^5 a^{-3} -3 z^5 a^{-5} +4 a^4 z^4+11 a^2 z^4+17 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} +15 z^4-6 a^3 z^3-21 a z^3-27 z^3 a^{-1} -10 z^3 a^{-3} +2 z^3 a^{-5} -5 a^4 z^2-17 a^2 z^2-20 z^2 a^{-2} -6 z^2 a^{-4} +z^2 a^{-6} -25 z^2+4 a^3 z+16 a z+18 z a^{-1} +6 z a^{-3} +2 a^4+8 a^2+8 a^{-2} +2 a^{-4} +13-2 a^3 z^{-1} -7 a z^{-1} -7 a^{-1} z^{-1} -2 a^{-3} z^{-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 \
-5-4-3-2-1012345χ
12          1-1
10         2 2
8        51 -4
6       52  3
4      65   -1
2     75    2
0    58     3
-2   45      -1
-4  15       4
-6 14        -3
-8 1         1
-101          -1
Integral Khovanov Homology

(db, data source)

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

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L10a38

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L10a40