L11n123

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L11n122.gif

L11n122

L11n124.gif

L11n124

Contents

L11n123.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n123 at Knotilus!


Link Presentations

[edit Notes on L11n123's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X9,16,10,17 X20,11,21,12 X8,21,9,22 X18,7,19,8 X12,19,13,20 X15,10,16,11 X22,17,5,18 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, 6, -5, -3, 8, 4, -7, 11, -2, -8, 3, 9, -6, 7, -4, 5, -9}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gif
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A Morse Link Presentation L11n123 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^5+3 t(1) t(2)^4-3 t(2)^4-4 t(1) t(2)^3+4 t(2)^3+4 t(1) t(2)^2-4 t(2)^2-3 t(1) t(2)+3 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{3}{q^{7/2}}-\frac{7}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{11}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{9}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{1}{q^{23/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z a^{11}-2 a^{11} z^{-1} +4 z^3 a^9+10 z a^9+5 a^9 z^{-1} -3 z^5 a^7-11 z^3 a^7-11 z a^7-3 a^7 z^{-1} -z^5 a^5-2 z^3 a^5 (db)
Kauffman polynomial -z^4 a^{14}+2 z^2 a^{14}-a^{14}-3 z^5 a^{13}+3 z^3 a^{13}-z^8 a^{12}+z^6 a^{12}-6 z^4 a^{12}+6 z^2 a^{12}-z^9 a^{11}-z^5 a^{11}-2 z^3 a^{11}+4 z a^{11}-2 a^{11} z^{-1} -5 z^8 a^{10}+10 z^6 a^{10}-8 z^4 a^{10}-4 z^2 a^{10}+5 a^{10}-z^9 a^9-6 z^7 a^9+20 z^5 a^9-25 z^3 a^9+15 z a^9-5 a^9 z^{-1} -4 z^8 a^8+6 z^6 a^8+2 z^4 a^8-8 z^2 a^8+5 a^8-6 z^7 a^7+17 z^5 a^7-18 z^3 a^7+11 z a^7-3 a^7 z^{-1} -3 z^6 a^6+5 z^4 a^6-z^5 a^5+2 z^3 a^5 (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
-4         11
-6        31-2
-8       4  4
-10      43  -1
-12     74   3
-14    45    1
-16   56     -1
-18  24      2
-20 15       -4
-22 2        2
-241         -1
Integral Khovanov Homology

(db, data source)

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