L11n163

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L11n162.gif

L11n162

L11n164.gif

L11n164

Contents

L11n163.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n163's Link Presentations]

Planar diagram presentation X8192 X10,3,11,4 X17,13,18,12 X14,5,15,6 X4,13,5,14 X11,19,12,18 X22,19,7,20 X20,15,21,16 X16,21,17,22 X2738 X6,9,1,10
Gauss code {1, -10, 2, -5, 4, -11}, {10, -1, 11, -2, -6, 3, 5, -4, 8, -9, -3, 6, 7, -8, 9, -7}
A Braid Representative
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A Morse Link Presentation L11n163 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^4-t(2)^4+t(1)^2 t(2)^3-4 t(1) t(2)^3+2 t(2)^3-3 t(1)^2 t(2)^2+5 t(1) t(2)^2-3 t(2)^2+2 t(1)^2 t(2)-4 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1) t(2)^2} (db)
Jones polynomial -\frac{2}{q^{3/2}}+\frac{4}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{9}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{5}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{1}{q^{21/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^9-z a^9+z^5 a^7+2 z^3 a^7+2 z a^7+a^7 z^{-1} +z^5 a^5+z^3 a^5-z a^5-a^5 z^{-1} -2 z^3 a^3-3 z a^3 (db)
Kauffman polynomial -z^6 a^{12}+3 z^4 a^{12}-2 z^2 a^{12}-3 z^7 a^{11}+10 z^5 a^{11}-9 z^3 a^{11}+2 z a^{11}-3 z^8 a^{10}+7 z^6 a^{10}-z^4 a^{10}-2 z^2 a^{10}-z^9 a^9-4 z^7 a^9+17 z^5 a^9-14 z^3 a^9+5 z a^9-5 z^8 a^8+9 z^6 a^8-z^4 a^8-z^2 a^8-z^9 a^7-3 z^7 a^7+7 z^5 a^7-2 z^3 a^7-2 z a^7+a^7 z^{-1} -2 z^8 a^6+z^4 a^6-a^6-2 z^7 a^5-2 z a^5+a^5 z^{-1} -z^6 a^4-2 z^4 a^4+z^2 a^4-3 z^3 a^3+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         22
-4        31-2
-6       41 3
-8      53  -2
-10     54   1
-12    45    1
-14   45     -1
-16  25      3
-18 13       -2
-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}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}\oplus{\mathbb Z}_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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