L11n150

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L11n149.gif

L11n149

L11n151.gif

L11n151

Contents

L11n150.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n150 at Knotilus!


Link Presentations

[edit Notes on L11n150's Link Presentations]

Planar diagram presentation X8192 X18,9,19,10 X6718 X22,19,7,20 X12,5,13,6 X3,10,4,11 X15,5,16,4 X11,16,12,17 X20,13,21,14 X14,21,15,22 X2,18,3,17
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.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n150 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^2-u^2 v-3 u v^2+5 u v-3 u-v+1}{u v} (db)
Jones polynomial -\frac{1}{q^{3/2}}+\frac{1}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{4}{q^{9/2}}-\frac{5}{q^{11/2}}+\frac{5}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^9-2 z a^9-a^9 z^{-1} +z^5 a^7+4 z^3 a^7+6 z a^7+2 a^7 z^{-1} -2 z^3 a^5-3 z a^5-z^3 a^3-2 z a^3-a^3 z^{-1} (db)
Kauffman polynomial -z^6 a^{12}+4 z^4 a^{12}-3 z^2 a^{12}-2 z^7 a^{11}+8 z^5 a^{11}-7 z^3 a^{11}+z a^{11}-2 z^8 a^{10}+8 z^6 a^{10}-9 z^4 a^{10}+6 z^2 a^{10}-2 a^{10}-z^9 a^9+3 z^7 a^9-2 z^5 a^9+2 z^3 a^9-2 z a^9+a^9 z^{-1} -3 z^8 a^8+14 z^6 a^8-25 z^4 a^8+20 z^2 a^8-5 a^8-z^9 a^7+5 z^7 a^7-12 z^5 a^7+13 z^3 a^7-8 z a^7+2 a^7 z^{-1} -z^8 a^6+5 z^6 a^6-13 z^4 a^6+11 z^2 a^6-3 a^6-2 z^5 a^5+3 z^3 a^5-3 z a^5-z^4 a^4+a^4-z^3 a^3+2 z a^3-a^3 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
-2         11
-4        110
-6       3  3
-8      22  0
-10     32   1
-12    22    0
-14   23     -1
-16  12      1
-18 12       -1
-20 1        1
-221         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-4 i=-2
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}^{2}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
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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L11n149.gif

L11n149

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L11n151