L11n149

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L11n148.gif

L11n148

L11n150.gif

L11n150

Contents

L11n149.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n149 at Knotilus!


Link Presentations

[edit Notes on L11n149's Link Presentations]

Planar diagram presentation X8192 X9,19,10,18 X6718 X22,19,7,20 X12,5,13,6 X3,10,4,11 X4,15,5,16 X16,12,17,11 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.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart2.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 L11n149 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^4-t(2)^4-3 t(1) t(2)^3+2 t(2)^3-2 t(1)^2 t(2)^2+3 t(1) t(2)^2-2 t(2)^2+2 t(1)^2 t(2)-3 t(1) t(2)-t(1)^2+t(1)}{t(1) t(2)^2} (db)
Jones polynomial \frac{6}{q^{9/2}}-\frac{5}{q^{7/2}}+\frac{2}{q^{5/2}}-\frac{1}{q^{3/2}}+\frac{1}{q^{21/2}}-\frac{3}{q^{19/2}}+\frac{4}{q^{17/2}}-\frac{6}{q^{15/2}}+\frac{7}{q^{13/2}}-\frac{7}{q^{11/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 \left(-z^3\right)-a^9 z+a^7 z^5+3 a^7 z^3+4 a^7 z+a^7 z^{-1} -3 a^5 z^3-5 a^5 z-a^5 z^{-1} -a^3 z^3-a^3 z (db)
Kauffman polynomial -z^6 a^{12}+3 z^4 a^{12}-z^2 a^{12}-3 z^7 a^{11}+11 z^5 a^{11}-9 z^3 a^{11}+z a^{11}-3 z^8 a^{10}+10 z^6 a^{10}-7 z^4 a^{10}+z^2 a^{10}-z^9 a^9-z^7 a^9+12 z^5 a^9-13 z^3 a^9+4 z a^9-4 z^8 a^8+13 z^6 a^8-14 z^4 a^8+6 z^2 a^8-z^9 a^7+2 z^7 a^7-3 z^5 a^7+3 z^3 a^7-3 z a^7+a^7 z^{-1} -z^8 a^6+2 z^6 a^6-6 z^4 a^6+5 z^2 a^6-a^6-4 z^5 a^5+6 z^3 a^5-5 z a^5+a^5 z^{-1} -2 z^4 a^4+z^2 a^4-z^3 a^3+z a^3 (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        21-1
-6       3  3
-8      32  -1
-10     43   1
-12    33    0
-14   34     -1
-16  24      2
-18 12       -1
-20 2        2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

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L11n148

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