L10a119

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L10a118.gif

L10a118

L10a120.gif

L10a120

Contents

L10a119.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a119 at Knotilus!


Link Presentations

[edit Notes on L10a119's Link Presentations]

Planar diagram presentation X12,1,13,2 X20,7,11,8 X14,3,15,4 X6,15,7,16 X16,5,17,6 X4,17,5,18 X18,9,19,10 X2,11,3,12 X10,13,1,14 X8,19,9,20
Gauss code {1, -8, 3, -6, 5, -4, 2, -10, 7, -9}, {8, -1, 9, -3, 4, -5, 6, -7, 10, -2}
A Braid Representative
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A Morse Link Presentation L10a119 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^4 (-v)-3 u^3 v^2+3 u^3 v-u^3-3 u^2 v^3+5 u^2 v^2-3 u^2 v-u v^4+3 u v^3-3 u v^2-v^3}{u^2 v^2} (db)
Jones polynomial \frac{7}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{3/2}}-\frac{1}{q^{23/2}}+\frac{1}{q^{21/2}}-\frac{4}{q^{19/2}}+\frac{6}{q^{17/2}}-\frac{7}{q^{15/2}}+\frac{9}{q^{13/2}}-\frac{9}{q^{11/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z a^{11}+a^{11} z^{-1} -z^3 a^9-a^9 z^{-1} -3 z^3 a^7-3 z a^7-3 z^3 a^5-3 z a^5-z^3 a^3 (db)
Kauffman polynomial a^{13} z^7-6 a^{13} z^5+12 a^{13} z^3-8 a^{13} z+a^{12} z^8-3 a^{12} z^6+4 a^{12} z^2+a^{11} z^9-2 a^{11} z^7-2 a^{11} z^5+4 a^{11} z^3-a^{11} z^{-1} +4 a^{10} z^8-10 a^{10} z^6+7 a^{10} z^2+a^{10}+a^9 z^9+3 a^9 z^7-11 a^9 z^5+2 a^9 z^3+2 a^9 z-a^9 z^{-1} +3 a^8 z^8-12 a^8 z^4+6 a^8 z^2+6 a^7 z^7-9 a^7 z^5+3 a^7 z^3-3 a^7 z+7 a^6 z^6-9 a^6 z^4+3 a^6 z^2+6 a^5 z^5-6 a^5 z^3+3 a^5 z+3 a^4 z^4+a^3 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 \
-10-9-8-7-6-5-4-3-2-10χ
-2          11
-4         31-2
-6        3  3
-8       43  -1
-10      53   2
-12     44    0
-14    35     -2
-16   34      1
-18  13       -2
-20  3        3
-2211         0
-241          1
Integral Khovanov Homology

(db, data source)

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