L11n116

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L11n115.gif

L11n115

L11n117.gif

L11n117

Contents

L11n116.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11n116's Link Presentations]

Planar diagram presentation X6172 X7,17,8,16 X20,17,21,18 X18,13,19,14 X14,19,15,20 X4,21,1,22 X10,5,11,6 X12,3,13,4 X22,11,5,12 X2,9,3,10 X15,9,16,8
Gauss code {1, -10, 8, -6}, {7, -1, -2, 11, 10, -7, 9, -8, 4, -5, -11, 2, 3, -4, 5, -3, 6, -9}
A Braid Representative
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A Morse Link Presentation L11n116 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(2)^5+2 t(1) t(2)^4-4 t(2)^4-7 t(1) t(2)^3+7 t(2)^3+7 t(1) t(2)^2-7 t(2)^2-4 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{3}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{11}{q^{7/2}}+\frac{13}{q^{9/2}}-\frac{14}{q^{11/2}}+\frac{14}{q^{13/2}}-\frac{11}{q^{15/2}}+\frac{7}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{1}{q^{21/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 \left(-z^3\right)+a^9 z^{-1} +a^7 z^5-3 a^7 z-3 a^7 z^{-1} +2 a^5 z^5+5 a^5 z^3+6 a^5 z+4 a^5 z^{-1} -3 a^3 z^3-5 a^3 z-2 a^3 z^{-1} (db)
Kauffman polynomial a^{12} z^6-2 a^{12} z^4+a^{12} z^2+4 a^{11} z^7-11 a^{11} z^5+8 a^{11} z^3+5 a^{10} z^8-11 a^{10} z^6+4 a^{10} z^4+a^{10} z^2-a^{10}+2 a^9 z^9+7 a^9 z^7-28 a^9 z^5+19 a^9 z^3-4 a^9 z+a^9 z^{-1} +11 a^8 z^8-22 a^8 z^6+7 a^8 z^4+2 a^8 z^2-3 a^8+2 a^7 z^9+10 a^7 z^7-31 a^7 z^5+27 a^7 z^3-12 a^7 z+3 a^7 z^{-1} +6 a^6 z^8-7 a^6 z^6+a^6 z^4+5 a^6 z^2-3 a^6+7 a^5 z^7-14 a^5 z^5+22 a^5 z^3-15 a^5 z+4 a^5 z^{-1} +3 a^4 z^6+3 a^4 z^2-2 a^4+6 a^3 z^3-7 a^3 z+2 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         33
-4        41-3
-6       72 5
-8      64  -2
-10     87   1
-12    77    0
-14   47     -3
-16  37      4
-18 14       -3
-20 3        3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}

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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L11n115.gif

L11n115

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L11n117