L11n104

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L11n103.gif

L11n103

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L11n105

Contents

L11n104.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n104 at Knotilus!


Link Presentations

[edit Notes on L11n104's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X7,16,8,17 X22,18,5,17 X18,14,19,13 X21,14,22,15 X9,20,10,21 X15,8,16,9 X19,10,20,11 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, -3, 8, -7, 9, 11, -2, 5, 6, -8, 3, 4, -5, -9, 7, -6, -4}
A Braid Representative
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A Morse Link Presentation L11n104 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^3-2 t(2)^3-6 t(1) t(2)^2+6 t(2)^2+6 t(1) t(2)-6 t(2)-2 t(1)+1}{\sqrt{t(1)} t(2)^{3/2}} (db)
Jones polynomial \frac{10}{q^{9/2}}-\frac{11}{q^{7/2}}+\frac{8}{q^{5/2}}-\frac{7}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{9}{q^{11/2}}-\sqrt{q}+\frac{4}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^9 z^{-1} +4 z a^7+3 a^7 z^{-1} -4 z^3 a^5-6 z a^5-2 a^5 z^{-1} +z^5 a^3+z^3 a^3-z^3 a (db)
Kauffman polynomial a^{10} z^4-2 a^{10} z^2+a^{10}+3 a^9 z^5-4 a^9 z^3+a^9 z-a^9 z^{-1} +a^8 z^8-2 a^8 z^6+9 a^8 z^4-10 a^8 z^2+3 a^8+a^7 z^9-a^7 z^7+5 a^7 z^5-6 a^7 z^3+7 a^7 z-3 a^7 z^{-1} +5 a^6 z^8-12 a^6 z^6+17 a^6 z^4-10 a^6 z^2+3 a^6+a^5 z^9+5 a^5 z^7-12 a^5 z^5+4 a^5 z^3+6 a^5 z-2 a^5 z^{-1} +4 a^4 z^8-6 a^4 z^6+2 a^4 z^4-2 a^4 z^2+6 a^3 z^7-13 a^3 z^5+5 a^3 z^3+4 a^2 z^6-7 a^2 z^4+a z^5-a z^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 \
-7-6-5-4-3-2-1012χ
2         11
0        3 -3
-2       41 3
-4      54  -1
-6     63   3
-8    45    1
-10   56     -1
-12  25      3
-14 14       -3
-16 2        2
-181         -1
Integral Khovanov Homology

(db, data source)

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

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