L11a235

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L11a234

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L11a236

Contents

L11a235.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L11a235's Link Presentations]

Planar diagram presentation X8192 X12,3,13,4 X20,13,21,14 X16,9,17,10 X10,19,11,20 X22,15,7,16 X14,21,15,22 X18,5,19,6 X2738 X4,11,5,12 X6,17,1,18
Gauss code {1, -9, 2, -10, 8, -11}, {9, -1, 4, -5, 10, -2, 3, -7, 6, -4, 11, -8, 5, -3, 7, -6}
A Braid Representative
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A Morse Link Presentation L11a235 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(2)^6-3 t(1) t(2)^5+3 t(2)^5-3 t(1)^2 t(2)^4+6 t(1) t(2)^4-4 t(2)^4+4 t(1)^2 t(2)^3-7 t(1) t(2)^3+4 t(2)^3-4 t(1)^2 t(2)^2+6 t(1) t(2)^2-3 t(2)^2+3 t(1)^2 t(2)-3 t(1) t(2)-t(1)^2}{t(1) t(2)^3} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{11}{q^{11/2}}-\frac{16}{q^{13/2}}+\frac{17}{q^{15/2}}-\frac{18}{q^{17/2}}+\frac{15}{q^{19/2}}-\frac{11}{q^{21/2}}+\frac{7}{q^{23/2}}-\frac{3}{q^{25/2}}+\frac{1}{q^{27/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^{13} (-z)-2 a^{13} z^{-1} +4 a^{11} z^3+10 a^{11} z+5 a^{11} z^{-1} -3 a^9 z^5-9 a^9 z^3-8 a^9 z-3 a^9 z^{-1} -3 a^7 z^5-8 a^7 z^3-4 a^7 z-a^5 z^5-2 a^5 z^3 (db)
Kauffman polynomial a^{16} z^6-3 a^{16} z^4+3 a^{16} z^2-a^{16}+3 a^{15} z^7-8 a^{15} z^5+5 a^{15} z^3+4 a^{14} z^8-7 a^{14} z^6-2 a^{14} z^4+4 a^{14} z^2+4 a^{13} z^9-7 a^{13} z^7+5 a^{13} z^5-10 a^{13} z^3+7 a^{13} z-2 a^{13} z^{-1} +2 a^{12} z^{10}+2 a^{12} z^8-12 a^{12} z^6+17 a^{12} z^4-17 a^{12} z^2+5 a^{12}+10 a^{11} z^9-31 a^{11} z^7+51 a^{11} z^5-47 a^{11} z^3+22 a^{11} z-5 a^{11} z^{-1} +2 a^{10} z^{10}+5 a^{10} z^8-23 a^{10} z^6+36 a^{10} z^4-21 a^{10} z^2+5 a^{10}+6 a^9 z^9-15 a^9 z^7+23 a^9 z^5-18 a^9 z^3+11 a^9 z-3 a^9 z^{-1} +7 a^8 z^8-16 a^8 z^6+15 a^8 z^4-3 a^8 z^2+6 a^7 z^7-14 a^7 z^5+12 a^7 z^3-4 a^7 z+3 a^6 z^6-5 a^6 z^4+a^5 z^5-2 a^5 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 \
-11-10-9-8-7-6-5-4-3-2-10χ
-4           11
-6          31-2
-8         4  4
-10        73  -4
-12       94   5
-14      87    -1
-16     109     1
-18    69      3
-20   59       -4
-22  26        4
-24 15         -4
-26 2          2
-281           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-8 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-7 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-6 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=-5 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
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

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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L11a234

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L11a236