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

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Link Presentations

[edit Notes on L10a122's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,12,17,11 X18,13,19,14 X20,18,9,17 X12,19,13,20 X8,16,5,15 X14,8,15,7 X2536 X4,9,1,10
Gauss code {1, -9, 2, -10}, {9, -1, 8, -7}, {10, -2, 3, -6, 4, -8, 7, -3, 5, -4, 6, -5}
A Braid Representative
A Morse Link Presentation L10a122 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{-t(1) t(3)^3+t(1) t(2) t(3)^3-2 t(2) t(3)^3+t(3)^3+4 t(1) t(3)^2-3 t(1) t(2) t(3)^2+6 t(2) t(3)^2-3 t(3)^2-6 t(1) t(3)+3 t(1) t(2) t(3)-4 t(2) t(3)+3 t(3)+2 t(1)-t(1) t(2)+t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial  q^{-6} -2 q^{-5} +q^4+6 q^{-4} -4 q^3-9 q^{-3} +9 q^2+13 q^{-2} -11 q-14 q^{-1} +14 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 z^{-2} +a^6-3 z^2 a^4-3 a^4 z^{-2} -5 a^4+3 z^4 a^2+7 z^2 a^2+4 a^2 z^{-2} +8 a^2-z^6-3 z^4-6 z^2-3 z^{-2} -6+z^4 a^{-2} +z^2 a^{-2} + a^{-2} z^{-2} +2 a^{-2} (db)
Kauffman polynomial a^6 z^6-4 a^6 z^4+6 a^6 z^2+a^6 z^{-2} -4 a^6+2 a^5 z^7-5 a^5 z^5+3 a^5 z^3+a^5 z-a^5 z^{-1} +2 a^4 z^8+a^4 z^6-16 a^4 z^4+z^4 a^{-4} +24 a^4 z^2+3 a^4 z^{-2} -14 a^4+a^3 z^9+7 a^3 z^7-20 a^3 z^5+4 z^5 a^{-3} +12 a^3 z^3-z^3 a^{-3} +a^3 z-a^3 z^{-1} +7 a^2 z^8-2 a^2 z^6+9 z^6 a^{-2} -30 a^2 z^4-11 z^4 a^{-2} +40 a^2 z^2+6 z^2 a^{-2} +4 a^2 z^{-2} + a^{-2} z^{-2} -21 a^2-4 a^{-2} +a z^9+15 a z^7+10 z^7 a^{-1} -32 a z^5-13 z^5 a^{-1} +14 a z^3+4 z^3 a^{-1} +a z+z a^{-1} -a z^{-1} - a^{-1} z^{-1} +5 z^8+7 z^6-30 z^4+28 z^2+3 z^{-2} -14 (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 \
9          11
7         3 -3
5        61 5
3       53  -2
1      96   3
-1     88    0
-3    56     -1
-5   48      4
-7  25       -3
-9 15        4
-11 1         -1
-131          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{9}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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See/edit the Link Page master template (intermediate).

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