L10a49

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L10a48.gif

L10a48

L10a50.gif

L10a50

Contents

L10a49.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

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Link Presentations

[edit Notes on L10a49's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X10,5,11,6 X14,3,15,4 X18,11,19,12 X20,13,5,14 X12,19,13,20 X2,9,3,10 X8,15,9,16
Gauss code {1, -9, 5, -3}, {4, -1, 2, -10, 9, -4, 6, -8, 7, -5, 10, -2, 3, -6, 8, -7}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation L10a49 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^5+4 t(1) t(2)^4-4 t(2)^4-6 t(1) t(2)^3+6 t(2)^3+6 t(1) t(2)^2-6 t(2)^2-4 t(1) t(2)+4 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{7}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{1}{q^{5/2}}-\frac{1}{q^{25/2}}+\frac{4}{q^{23/2}}-\frac{7}{q^{21/2}}+\frac{11}{q^{19/2}}-\frac{13}{q^{17/2}}+\frac{14}{q^{15/2}}-\frac{14}{q^{13/2}}+\frac{9}{q^{11/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial z^3 a^{11}-2 a^{11} z^{-1} -z^5 a^9+z^3 a^9+8 z a^9+5 a^9 z^{-1} -3 z^5 a^7-10 z^3 a^7-10 z a^7-3 a^7 z^{-1} -z^5 a^5-2 z^3 a^5 (db)
Kauffman polynomial -z^5 a^{15}+z^3 a^{15}-4 z^6 a^{14}+7 z^4 a^{14}-2 z^2 a^{14}-a^{14}-6 z^7 a^{13}+10 z^5 a^{13}-3 z^3 a^{13}+z a^{13}-5 z^8 a^{12}+5 z^6 a^{12}+z^4 a^{12}+z^2 a^{12}-2 z^9 a^{11}-6 z^7 a^{11}+16 z^5 a^{11}-11 z^3 a^{11}+5 z a^{11}-2 a^{11} z^{-1} -10 z^8 a^{10}+20 z^6 a^{10}-13 z^4 a^{10}-4 z^2 a^{10}+5 a^{10}-2 z^9 a^9-6 z^7 a^9+22 z^5 a^9-27 z^3 a^9+14 z a^9-5 a^9 z^{-1} -5 z^8 a^8+8 z^6 a^8-2 z^4 a^8-7 z^2 a^8+5 a^8-6 z^7 a^7+16 z^5 a^7-18 z^3 a^7+10 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 \
-10-9-8-7-6-5-4-3-2-10χ
-4          11
-6         31-2
-8        4  4
-10       53  -2
-12      94   5
-14     66    0
-16    78     -1
-18   46      2
-20  37       -4
-22 14        3
-24 3         -3
-261          1
Integral Khovanov Homology

(db, data source)

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