L10a101

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L10a100.gif

L10a100

L10a102.gif

L10a102

Contents

L10a101.gif
(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L10a101 at Knotilus!

Contains two L4a1 configurations.

As a Celtic (or pseudo-Celtic) linear decorative knot
(multicolored)
Mongolian ornament
Rotated knotwork cross with four L10a101 sub-configurations

Link Presentations

[edit Notes on L10a101's Link Presentations]

Planar diagram presentation X10,1,11,2 X12,3,13,4 X14,19,15,20 X18,7,19,8 X6,15,7,16 X16,5,17,6 X4,17,5,18 X20,13,9,14 X2,9,3,10 X8,11,1,12
Gauss code {1, -9, 2, -7, 6, -5, 4, -10}, {9, -1, 10, -2, 8, -3, 5, -6, 7, -4, 3, -8}
A Braid Representative
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A Morse Link Presentation L10a101 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)+t(2)) \left(2 t(2) t(1)^2-t(1)^2+2 t(2)^2 t(1)-4 t(2) t(1)+2 t(1)-t(2)^2+2 t(2)\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{9}{q^{11/2}}+\frac{9}{q^{13/2}}-\frac{8}{q^{15/2}}+\frac{6}{q^{17/2}}-\frac{4}{q^{19/2}}+\frac{2}{q^{21/2}}-\frac{1}{q^{23/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial z a^{11}-z^3 a^9+z a^9+a^9 z^{-1} -3 z^3 a^7-3 z a^7-a^7 z^{-1} -3 z^3 a^5-3 z a^5-z^3 a^3 (db)
Kauffman polynomial a^{13} z^7-5 a^{13} z^5+8 a^{13} z^3-4 a^{13} z+2 a^{12} z^8-9 a^{12} z^6+12 a^{12} z^4-4 a^{12} z^2+a^{11} z^9-11 a^{11} z^5+15 a^{11} z^3-3 a^{11} z+5 a^{10} z^8-15 a^{10} z^6+9 a^{10} z^4+a^{10} z^2+a^9 z^9+5 a^9 z^7-21 a^9 z^5+18 a^9 z^3-7 a^9 z+a^9 z^{-1} +3 a^8 z^8+a^8 z^6-15 a^8 z^4+8 a^8 z^2-a^8+6 a^7 z^7-9 a^7 z^5+4 a^7 z^3-5 a^7 z+a^7 z^{-1} +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    45     -1
-16   24      2
-18  24       -2
-20 13        2
-22 1         -1
-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}
r=-9 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-8 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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L10a100.gif

L10a100

L10a102.gif

L10a102