L10a11

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L10a10.gif

L10a10

L10a12.gif

L10a12

Contents

L10a11.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a11's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,11,17,12 X14,7,15,8 X8,15,9,16 X20,17,5,18 X18,14,19,13 X12,20,13,19 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {9, -1, 4, -5, 10, -2, 3, -8, 7, -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.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
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A Morse Link Presentation L10a11 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u v^4-7 u v^3+7 u v^2-4 u v+u+v^5-4 v^4+7 v^3-7 v^2+2 v}{\sqrt{u} v^{5/2}} (db)
Jones polynomial \frac{14}{q^{9/2}}-\frac{14}{q^{7/2}}+\frac{10}{q^{5/2}}-\frac{7}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{3}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{11}{q^{13/2}}-\frac{13}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z a^9+a^9 z^{-1} -3 z^3 a^7-5 z a^7-3 a^7 z^{-1} +2 z^5 a^5+5 z^3 a^5+6 z a^5+4 a^5 z^{-1} +z^5 a^3-3 z a^3-2 a^3 z^{-1} -z^3 a-z a (db)
Kauffman polynomial -z^5 a^{11}+2 z^3 a^{11}-z a^{11}-3 z^6 a^{10}+5 z^4 a^{10}-3 z^2 a^{10}+a^{10}-5 z^7 a^9+8 z^5 a^9-5 z^3 a^9+3 z a^9-a^9 z^{-1} -4 z^8 a^8+10 z^4 a^8-8 z^2 a^8+3 a^8-z^9 a^7-13 z^7 a^7+33 z^5 a^7-31 z^3 a^7+14 z a^7-3 a^7 z^{-1} -8 z^8 a^6+7 z^6 a^6+7 z^4 a^6-9 z^2 a^6+3 a^6-z^9 a^5-13 z^7 a^5+34 z^5 a^5-34 z^3 a^5+17 z a^5-4 a^5 z^{-1} -4 z^8 a^4+z^6 a^4+7 z^4 a^4-6 z^2 a^4+2 a^4-5 z^7 a^3+9 z^5 a^3-8 z^3 a^3+6 z a^3-2 a^3 z^{-1} -3 z^6 a^2+5 z^4 a^2-2 z^2 a^2-z^5 a+2 z^3 a-z a (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 \
-8-7-6-5-4-3-2-1012χ
2          11
0         2 -2
-2        51 4
-4       63  -3
-6      84   4
-8     66    0
-10    78     -1
-12   57      2
-14  26       -4
-16 15        4
-18 2         -2
-201          1
Integral Khovanov Homology

(db, data source)

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