L10a5

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L10a4.gif

L10a4

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L10a6

Contents

L10a5.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a5 at Knotilus!


Link Presentations

[edit Notes on L10a5's Link Presentations]

Planar diagram presentation X6172 X18,7,19,8 X4,19,1,20 X14,10,15,9 X8493 X12,5,13,6 X20,13,5,14 X16,12,17,11 X10,16,11,15 X2,18,3,17
Gauss code {1, -10, 5, -3}, {6, -1, 2, -5, 4, -9, 8, -6, 7, -4, 9, -8, 10, -2, 3, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gif
A Morse Link Presentation L10a5 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^4-3 v^3+3 v^2-3 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial -4 q^{9/2}+\frac{1}{q^{9/2}}+8 q^{7/2}-\frac{3}{q^{7/2}}-12 q^{5/2}+\frac{6}{q^{5/2}}+14 q^{3/2}-\frac{11}{q^{3/2}}+q^{11/2}-15 \sqrt{q}+\frac{13}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -a^3 z^3+6 a z^3-6 z^3 a^{-1} +2 z^3 a^{-3} -2 a^3 z+5 a z-4 z a^{-1} +z a^{-3} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -2 a z^9-2 z^9 a^{-1} -4 a^2 z^8-7 z^8 a^{-2} -11 z^8-3 a^3 z^7-5 a z^7-12 z^7 a^{-1} -10 z^7 a^{-3} -a^4 z^6+9 a^2 z^6+5 z^6 a^{-2} -8 z^6 a^{-4} +23 z^6+9 a^3 z^5+25 a z^5+34 z^5 a^{-1} +14 z^5 a^{-3} -4 z^5 a^{-5} +3 a^4 z^4-4 a^2 z^4+5 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} -11 z^4-9 a^3 z^3-26 a z^3-26 z^3 a^{-1} -7 z^3 a^{-3} +2 z^3 a^{-5} -2 a^4 z^2-a^2 z^2-3 z^2 a^{-2} -2 z^2 a^{-4} +4 a^3 z+10 a z+8 z a^{-1} +2 z a^{-3} +1-a z^{-1} - a^{-1} 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         3 3
8        51 -4
6       73  4
4      75   -2
2     87    1
0    79     2
-2   46      -2
-4  27       5
-6 14        -3
-8 2         2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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