L10a22

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L10a21

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L10a23

Contents

L10a22.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a22 at Knotilus!


Link Presentations

[edit Notes on L10a22's Link Presentations]

Planar diagram presentation X6172 X14,9,15,10 X4,19,1,20 X12,6,13,5 X10,4,11,3 X20,12,5,11 X2,14,3,13 X18,16,19,15 X16,8,17,7 X8,18,9,17
Gauss code {1, -7, 5, -3}, {4, -1, 9, -10, 2, -5, 6, -4, 7, -2, 8, -9, 10, -8, 3, -6}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation L10a22 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{17/2}-4 q^{15/2}+9 q^{13/2}-12 q^{11/2}+15 q^{9/2}-17 q^{7/2}+14 q^{5/2}-12 q^{3/2}+7 \sqrt{q}-\frac{4}{\sqrt{q}}+\frac{1}{q^{3/2}} (db)
Signature 3 (db)
HOMFLY-PT polynomial z^3 a^{-7} +z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -5 z^3 a^{-5} -4 z a^{-5} -2 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +6 z^3 a^{-3} +3 z a^{-3} -z^5 a^{-1} -2 z^3 a^{-1} + a^{-1} z^{-1} (db)
Kauffman polynomial z^4 a^{-10} +4 z^5 a^{-9} -z^3 a^{-9} +9 z^6 a^{-8} -10 z^4 a^{-8} +5 z^2 a^{-8} -2 a^{-8} +11 z^7 a^{-7} -15 z^5 a^{-7} +8 z^3 a^{-7} -2 z a^{-7} + a^{-7} z^{-1} +7 z^8 a^{-6} +z^6 a^{-6} -19 z^4 a^{-6} +14 z^2 a^{-6} -5 a^{-6} +2 z^9 a^{-5} +16 z^7 a^{-5} -41 z^5 a^{-5} +26 z^3 a^{-5} -7 z a^{-5} +2 a^{-5} z^{-1} +12 z^8 a^{-4} -19 z^6 a^{-4} -4 z^4 a^{-4} +11 z^2 a^{-4} -3 a^{-4} +2 z^9 a^{-3} +9 z^7 a^{-3} -33 z^5 a^{-3} +26 z^3 a^{-3} -6 z a^{-3} +5 z^8 a^{-2} -10 z^6 a^{-2} +2 z^4 a^{-2} +3 z^2 a^{-2} + a^{-2} +4 z^7 a^{-1} -11 z^5 a^{-1} +9 z^3 a^{-1} -z a^{-1} - a^{-1} z^{-1} +z^6-2 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 \
-3-2-101234567χ
18          1-1
16         3 3
14        61 -5
12       63  3
10      96   -3
8     86    2
6    69     3
4   68      -2
2  38       5
0 14        -3
-2 3         3
-41          -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=2 i=4
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\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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L10a21

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L10a23