L10a37

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L10a36.gif

L10a36

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L10a38

Contents

L10a37.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a37's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X18,8,19,7 X20,15,5,16 X16,19,17,20 X14,10,15,9 X10,14,11,13 X8,18,9,17 X2536 X4,11,1,12
Gauss code {1, -9, 2, -10}, {9, -1, 3, -8, 6, -7, 10, -2, 7, -6, 4, -5, 8, -3, 5, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation L10a37 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^3-2 t(2)^3-6 t(1) t(2)^2+8 t(2)^2+8 t(1) t(2)-6 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial -q^{7/2}+3 q^{5/2}-6 q^{3/2}+9 \sqrt{q}-\frac{11}{\sqrt{q}}+\frac{11}{q^{3/2}}-\frac{11}{q^{5/2}}+\frac{8}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 z^{-1} -3 z a^5-2 a^5 z^{-1} +3 z^3 a^3+3 z a^3+2 a^3 z^{-1} -z^5 a-z^3 a-2 z a-a z^{-1} +2 z^3 a^{-1} +z a^{-1} -z a^{-3} (db)
Kauffman polynomial a^7 z^5-3 a^7 z^3+3 a^7 z-a^7 z^{-1} +2 a^6 z^6-4 a^6 z^4+2 a^6 z^2+2 a^5 z^7+a^5 z^5-9 a^5 z^3+8 a^5 z-2 a^5 z^{-1} +2 a^4 z^8+a^4 z^6-5 a^4 z^4+4 a^4 z^2-a^4+a^3 z^9+3 a^3 z^7-a^3 z^5+z^5 a^{-3} -8 a^3 z^3-2 z^3 a^{-3} +8 a^3 z+z a^{-3} -2 a^3 z^{-1} +5 a^2 z^8-5 a^2 z^6+3 z^6 a^{-2} +a^2 z^4-6 z^4 a^{-2} +3 z^2 a^{-2} +a z^9+5 a z^7+4 z^7 a^{-1} -8 a z^5-6 z^5 a^{-1} +a z^3+z^3 a^{-1} +2 a z-a z^{-1} +3 z^8-z^6-4 z^4+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 \
-6-5-4-3-2-101234χ
8          11
6         2 -2
4        41 3
2       52  -3
0      64   2
-2     66    0
-4    55     0
-6   36      3
-8  25       -3
-10 14        3
-12 1         -1
-141          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\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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L10a36

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L10a38