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

Visit L10a54 at Knotilus!

Link Presentations

[edit Notes on L10a54's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X18,11,19,12 X12,6,13,5 X4,18,5,17 X14,7,15,8 X16,14,17,13 X20,15,7,16 X6,19,1,20
Gauss code {1, -2, 3, -6, 5, -10}, {7, -1, 2, -3, 4, -5, 8, -7, 9, -8, 6, -4, 10, -9}
A Braid Representative
A Morse Link Presentation L10a54 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^4-t(1) t(2)^4-4 t(1)^2 t(2)^3+5 t(1) t(2)^3-t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2-t(1)^2 t(2)+5 t(1) t(2)-4 t(2)-t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial q^{3/2}-4 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{10}{q^{3/2}}+\frac{12}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{12}{q^{9/2}}-\frac{10}{q^{11/2}}+\frac{6}{q^{13/2}}-\frac{3}{q^{15/2}}+\frac{1}{q^{17/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 \left(-z^3\right)-2 a^7 z+2 a^5 z^5+6 a^5 z^3+4 a^5 z-a^3 z^7-4 a^3 z^5-5 a^3 z^3-2 a^3 z+a^3 z^{-1} +a z^5+2 a z^3-a z-a z^{-1} (db)
Kauffman polynomial -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+3 z^3 a^9-z a^9-5 z^6 a^8+5 z^4 a^8-2 z^2 a^8-6 z^7 a^7+7 z^5 a^7-4 z^3 a^7+z a^7-5 z^8 a^6+5 z^6 a^6-z^4 a^6-2 z^9 a^5-6 z^7 a^5+21 z^5 a^5-18 z^3 a^5+5 z a^5-10 z^8 a^4+25 z^6 a^4-19 z^4 a^4+6 z^2 a^4-2 z^9 a^3-4 z^7 a^3+23 z^5 a^3-19 z^3 a^3+2 z a^3+a^3 z^{-1} -5 z^8 a^2+14 z^6 a^2-10 z^4 a^2+3 z^2 a^2-a^2-4 z^7 a+12 z^5 a-8 z^3 a-z a+a z^{-1} -z^6+2 z^4 (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 \
4          1-1
2         3 3
0        31 -2
-2       73  4
-4      64   -2
-6     76    1
-8    67     1
-10   46      -2
-12  26       4
-14 14        -3
-16 2         2
-181          -1
Integral Khovanov Homology

(db, data source)

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