L11n151

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L11n150.gif

L11n150

L11n152.gif

L11n152

Contents

L11n151.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n151 at Knotilus!


Link Presentations

[edit Notes on L11n151's Link Presentations]

Planar diagram presentation X8192 X18,9,19,10 X6718 X22,19,7,20 X12,5,13,6 X3,10,4,11 X4,15,5,16 X11,16,12,17 X20,13,21,14 X14,21,15,22 X17,2,18,3
Gauss code {1, 11, -6, -7, 5, -3}, {3, -1, 2, 6, -8, -5, 9, -10, 7, 8, -11, -2, 4, -9, 10, -4}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n151 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 u^2 v^4-2 u^2 v^3+u^2 v^2+u v^3-u v^2+u v+v^2-2 v+2}{u v^2} (db)
Jones polynomial \frac{1}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{25/2}}-\frac{2}{q^{23/2}}+\frac{3}{q^{21/2}}-\frac{4}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{4}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{3}{q^{11/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial -z a^{13}-a^{13} z^{-1} +z^5 a^{11}+5 z^3 a^{11}+6 z a^{11}+2 a^{11} z^{-1} -z^7 a^9-5 z^5 a^9-6 z^3 a^9-z a^9-z^7 a^7-6 z^5 a^7-11 z^3 a^7-7 z a^7-a^7 z^{-1} (db)
Kauffman polynomial -z^2 a^{16}-2 z^3 a^{15}+z a^{15}-3 z^4 a^{14}+4 z^2 a^{14}-2 a^{14}-z^7 a^{13}+2 z^5 a^{13}-z^3 a^{13}-z a^{13}+a^{13} z^{-1} -2 z^8 a^{12}+10 z^6 a^{12}-20 z^4 a^{12}+19 z^2 a^{12}-5 a^{12}-z^9 a^{11}+4 z^7 a^{11}-7 z^5 a^{11}+12 z^3 a^{11}-9 z a^{11}+2 a^{11} z^{-1} -3 z^8 a^{10}+14 z^6 a^{10}-19 z^4 a^{10}+11 z^2 a^{10}-3 a^{10}-z^9 a^9+4 z^7 a^9-3 z^5 a^9-z^8 a^8+4 z^6 a^8-2 z^4 a^8-3 z^2 a^8+a^8-z^7 a^7+6 z^5 a^7-11 z^3 a^7+7 z a^7-a^7 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 \
-9-8-7-6-5-4-3-2-10χ
-6         11
-8        110
-10       2  2
-12      11  0
-14     32   1
-16    11    0
-18   33     0
-20  12      1
-22 12       -1
-24 1        1
-261         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-8 i=-6
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z} {\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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L11n150

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L11n152