L11n251

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L11n250.gif

L11n250

L11n252.gif

L11n252

Contents

L11n251.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n251 at Knotilus!


Link Presentations

[edit Notes on L11n251's Link Presentations]

Planar diagram presentation X12,1,13,2 X2,13,3,14 X8394 X11,17,12,16 X14,8,15,7 X6,16,7,15 X17,11,18,22 X4,20,5,19 X18,6,19,5 X20,9,21,10 X10,21,1,22
Gauss code {1, -2, 3, -8, 9, -6, 5, -3, 10, -11}, {-4, -1, 2, -5, 6, 4, -7, -9, 8, -10, 11, 7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart3.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L11n251 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-t(2) t(1)^2+t(1)^2+3 t(2) t(1)+t(2)^2-t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{3}{q^{9/2}}-2 q^{7/2}+\frac{7}{q^{7/2}}+5 q^{5/2}-\frac{11}{q^{5/2}}-8 q^{3/2}+\frac{11}{q^{3/2}}+\frac{1}{q^{11/2}}+11 \sqrt{q}-\frac{13}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -a^3 z^5-3 a^3 z^3-4 a^3 z-z a^{-3} +a z^7+5 a z^5-z^5 a^{-1} +10 a z^3-2 z^3 a^{-1} +7 a z-2 z a^{-1} +a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^6 z^4-a^6 z^2+3 a^5 z^5-2 a^5 z^3+6 a^4 z^6-7 a^4 z^4+4 a^4 z^2+8 a^3 z^7-15 a^3 z^5+3 z^5 a^{-3} +16 a^3 z^3-6 z^3 a^{-3} -8 a^3 z+2 z a^{-3} +5 a^2 z^8+z^8 a^{-2} -3 a^2 z^6+2 z^6 a^{-2} -6 a^2 z^4-9 z^4 a^{-2} +7 a^2 z^2+6 z^2 a^{-2} +a z^9+z^9 a^{-1} +11 a z^7+3 z^7 a^{-1} -30 a z^5-9 z^5 a^{-1} +29 a z^3+5 z^3 a^{-1} -14 a z-4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +6 z^8-7 z^6-7 z^4+8 z^2-1 (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 \
-5-4-3-2-101234χ
8         22
6        3 -3
4       52 3
2      63  -3
0     75   2
-2    68    2
-4   55     0
-6  26      4
-8 15       -4
-10 2        2
-121         -1
Integral Khovanov Homology

(db, data source)

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

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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L11n250.gif

L11n250

L11n252.gif

L11n252