L11n125

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L11n124.gif

L11n124

L11n126.gif

L11n126

Contents

L11n125.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n125 at Knotilus!


Link Presentations

[edit Notes on L11n125's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X9,22,10,5 X7,19,8,18 X17,9,18,8 X19,13,20,12 X11,21,12,20 X15,10,16,11 X21,16,22,17 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, -4, 5, -3, 8, -7, 6, 11, -2, -8, 9, -5, 4, -6, 7, -9, 3}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n125 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^5-t(1) t(2)^4-3 t(2)^4+4 t(2)^3+4 t(1) t(2)^2-3 t(1) t(2)-t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{6}{q^{9/2}}+\frac{5}{q^{7/2}}-q^{5/2}-\frac{5}{q^{5/2}}+q^{3/2}+\frac{4}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{3}{q^{13/2}}+\frac{4}{q^{11/2}}-\frac{2}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7+2 z^3 a^5+3 z a^5+2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-8 z a^3-4 a^3 z^{-1} +z^5 a+6 z^3 a+7 z a+3 a z^{-1} -z^3 a^{-1} -3 z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^5 z^9-a^3 z^9-3 a^6 z^8-4 a^4 z^8-2 a^2 z^8-z^8-3 a^7 z^7-a^5 z^7+2 a^3 z^7-a z^7-z^7 a^{-1} -a^8 z^6+10 a^6 z^6+18 a^4 z^6+14 a^2 z^6+7 z^6+11 a^7 z^5+18 a^5 z^5+12 a^3 z^5+11 a z^5+6 z^5 a^{-1} +3 a^8 z^4-5 a^6 z^4-23 a^4 z^4-27 a^2 z^4-12 z^4-9 a^7 z^3-27 a^5 z^3-34 a^3 z^3-25 a z^3-9 z^3 a^{-1} -a^8 z^2+a^6 z^2+10 a^4 z^2+15 a^2 z^2+7 z^2+3 a^7 z+15 a^5 z+23 a^3 z+16 a z+5 z a^{-1} -a^6-2 a^4-3 a^2-1-2 a^5 z^{-1} -4 a^3 z^{-1} -3 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 \
-7-6-5-4-3-2-101234χ
6           11
4            0
2        111 -1
0       31   2
-2      321   -2
-4     331    1
-6    33      0
-8   331      1
-10  24        2
-12 12         -1
-14 2          2
-161           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z} {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}_2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z} {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}
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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L11n126