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

Visit L10a129 at Knotilus!

Link Presentations

[edit Notes on L10a129's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X16,9,17,10 X14,8,15,7 X18,15,19,16 X20,14,11,13 X10,17,5,18 X8,20,9,19 X2536 X4,11,1,12
Gauss code {1, -9, 2, -10}, {9, -1, 4, -8, 3, -7}, {10, -2, 6, -4, 5, -3, 7, -5, 8, -6}
A Braid Representative
A Morse Link Presentation L10a129 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(3)^2 t(2)^2-2 t(3)^2 t(2)^2+t(1) t(2)^2-2 t(1) t(3) t(2)^2+3 t(3) t(2)^2-t(2)^2-2 t(1) t(3)^2 t(2)+3 t(3)^2 t(2)-3 t(1) t(2)+6 t(1) t(3) t(2)-6 t(3) t(2)+2 t(2)+t(1) t(3)^2-t(3)^2+2 t(1)-3 t(1) t(3)+2 t(3)-1}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial -q^2+4 q-7+11 q^{-1} -13 q^{-2} +15 q^{-3} -12 q^{-4} +11 q^{-5} -6 q^{-6} +3 q^{-7} - q^{-8} (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^8+3 z^2 a^6+a^6 z^{-2} +3 a^6-3 z^4 a^4-6 z^2 a^4-2 a^4 z^{-2} -5 a^4+z^6 a^2+3 z^4 a^2+5 z^2 a^2+a^2 z^{-2} +3 a^2-z^4-z^2 (db)
Kauffman polynomial z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-6 z^4 a^8+5 z^2 a^8-2 a^8+4 z^7 a^7-4 z^5 a^7-2 z^3 a^7+3 z a^7+3 z^8 a^6+4 z^6 a^6-17 z^4 a^6+17 z^2 a^6+a^6 z^{-2} -8 a^6+z^9 a^5+10 z^7 a^5-18 z^5 a^5+5 z^3 a^5+5 z a^5-2 a^5 z^{-1} +7 z^8 a^4-z^6 a^4-19 z^4 a^4+20 z^2 a^4+2 a^4 z^{-2} -9 a^4+z^9 a^3+12 z^7 a^3-24 z^5 a^3+10 z^3 a^3+3 z a^3-2 a^3 z^{-1} +4 z^8 a^2+2 z^6 a^2-15 z^4 a^2+11 z^2 a^2+a^2 z^{-2} -4 a^2+6 z^7 a-10 z^5 a+4 z^3 a+4 z^6-7 z^4+3 z^2+z^5 a^{-1} -z^3 a^{-1} (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 \
5          1-1
3         3 3
1        41 -3
-1       73  4
-3      75   -2
-5     86    2
-7    69     3
-9   56      -1
-11  27       5
-13 14        -3
-15 2         2
-171          -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-3 i=-1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}_2 {\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

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