L10a116

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L10a115.gif

L10a115

L10a117.gif

L10a117

Contents

L10a116.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a116 at Knotilus!


Symmetric depiction

Link Presentations

[edit Notes on L10a116's Link Presentations]

Planar diagram presentation X12,1,13,2 X20,9,11,10 X14,3,15,4 X8,15,9,16 X16,5,17,6 X18,7,19,8 X6,17,7,18 X4,19,5,20 X2,11,3,12 X10,13,1,14
Gauss code {1, -9, 3, -8, 5, -7, 6, -4, 2, -10}, {9, -1, 10, -3, 4, -5, 7, -6, 8, -2}
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
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A Morse Link Presentation L10a116 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(2)^2 t(1)^4-2 t(2)^3 t(1)^3+3 t(2)^2 t(1)^3-2 t(2) t(1)^3-t(2)^4 t(1)^2+3 t(2)^3 t(1)^2-5 t(2)^2 t(1)^2+3 t(2) t(1)^2-t(1)^2-2 t(2)^3 t(1)+3 t(2)^2 t(1)-2 t(2) t(1)-t(2)^2}{t(1)^2 t(2)^2} (db)
Jones polynomial -\frac{1}{q^{5/2}}+\frac{2}{q^{7/2}}-\frac{5}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{9}{q^{13/2}}+\frac{10}{q^{15/2}}-\frac{9}{q^{17/2}}+\frac{7}{q^{19/2}}-\frac{5}{q^{21/2}}+\frac{2}{q^{23/2}}-\frac{1}{q^{25/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^{11} z^3+2 a^{11} z+a^{11} z^{-1} -a^9 z^5-2 a^9 z^3-a^9 z-a^9 z^{-1} -2 a^7 z^5-6 a^7 z^3-4 a^7 z-a^5 z^5-3 a^5 z^3-2 a^5 z (db)
Kauffman polynomial -z^5 a^{15}+3 z^3 a^{15}-2 z a^{15}-2 z^6 a^{14}+4 z^4 a^{14}-z^2 a^{14}-3 z^7 a^{13}+7 z^5 a^{13}-7 z^3 a^{13}+4 z a^{13}-2 z^8 a^{12}+z^6 a^{12}+2 z^4 a^{12}-z^2 a^{12}-z^9 a^{11}-z^7 a^{11}+3 z^3 a^{11}-3 z a^{11}+a^{11} z^{-1} -4 z^8 a^{10}+6 z^6 a^{10}-4 z^4 a^{10}-a^{10}-z^9 a^9-z^7 a^9+3 z^3 a^9-3 z a^9+a^9 z^{-1} -2 z^8 a^8+z^6 a^8+2 z^4 a^8-z^2 a^8-3 z^7 a^7+7 z^5 a^7-7 z^3 a^7+4 z a^7-2 z^6 a^6+4 z^4 a^6-z^2 a^6-z^5 a^5+3 z^3 a^5-2 z 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         21-1
-8        3  3
-10       42  -2
-12      53   2
-14     54    -1
-16    45     -1
-18   35      2
-20  24       -2
-22  3        3
-2412         -1
-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} {\mathbb Z}
r=-9 {\mathbb Z}^{2}
r=-8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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L10a115.gif

L10a115

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L10a117