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

Visit L11n412 at Knotilus!

Link Presentations

[edit Notes on L11n412's Link Presentations]

Planar diagram presentation X8192 X5,15,6,14 X10,3,11,4 X13,5,14,4 X2738 X6,9,1,10 X11,18,12,19 X17,12,18,7 X15,20,16,21 X19,22,20,13 X21,16,22,17
Gauss code {1, -5, 3, 4, -2, -6}, {5, -1, 6, -3, -7, 8}, {-4, 2, -9, 11, -8, 7, -10, 9, -11, 10}
A Braid Representative
A Morse Link Presentation L11n412 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(3)^3-2 t(1)^2 t(3)^2+t(1) t(2)^2 t(3)^2-2 t(2)^2 t(3)^2+t(1) t(3)^2+t(1)^2 t(2) t(3)^2-2 t(1) t(2) t(3)^2+t(2) t(3)^2+2 t(1)^2 t(3)-t(1) t(2)^2 t(3)+2 t(2)^2 t(3)-t(1) t(3)-t(1)^2 t(2) t(3)+2 t(1) t(2) t(3)-t(2) t(3)-t(2)^2}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial - q^{-10} +2 q^{-9} -4 q^{-8} +6 q^{-7} -6 q^{-6} +8 q^{-5} -6 q^{-4} +6 q^{-3} -3 q^{-2} +2 q^{-1} (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^{10} z^{-2} -a^{10}+3 z^2 a^8+4 a^8 z^{-2} +6 a^8-2 z^4 a^6-6 z^2 a^6-5 a^6 z^{-2} -9 a^6-z^4 a^4+2 a^4 z^{-2} +2 a^4+2 z^2 a^2+2 a^2 (db)
Kauffman polynomial a^{11} z^7-5 a^{11} z^5+8 a^{11} z^3-5 a^{11} z+a^{11} z^{-1} +2 a^{10} z^8-9 a^{10} z^6+12 a^{10} z^4-7 a^{10} z^2-a^{10} z^{-2} +4 a^{10}+a^9 z^9+a^9 z^7-20 a^9 z^5+35 a^9 z^3-21 a^9 z+5 a^9 z^{-1} +6 a^8 z^8-26 a^8 z^6+38 a^8 z^4-32 a^8 z^2-4 a^8 z^{-2} +17 a^8+a^7 z^9+4 a^7 z^7-28 a^7 z^5+44 a^7 z^3-33 a^7 z+9 a^7 z^{-1} +4 a^6 z^8-15 a^6 z^6+26 a^6 z^4-32 a^6 z^2-5 a^6 z^{-2} +20 a^6+4 a^5 z^7-12 a^5 z^5+18 a^5 z^3-16 a^5 z+5 a^5 z^{-1} +2 a^4 z^6-4 a^4 z^2-2 a^4 z^{-2} +6 a^4+a^3 z^5+a^3 z^3+a^3 z+3 a^2 z^2-2 a^2 (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 \
-1         22
-3        32-1
-5       3  3
-7      44  0
-9     42   2
-11    24    2
-13   44     0
-15  13      2
-17 13       -2
-19 1        1
-211         -1
Integral Khovanov Homology

(db, data source)

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

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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