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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 L10a159's Link Presentations]

Planar diagram presentation X8192 X16,5,17,6 X14,3,15,4 X4,15,5,16 X12,17,7,18 X10,19,11,20 X18,9,19,10 X20,11,13,12 X2738 X6,13,1,14
Gauss code {1, -9, 3, -4, 2, -10}, {9, -1, 7, -6, 8, -5}, {10, -3, 4, -2, 5, -7, 6, -8}
A Braid Representative
A Morse Link Presentation L10a159 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^2 t(3)^3+t(1)^2 t(2) t(3)^3-t(1) t(2) t(3)^3+t(1)^2 t(3)^2+t(1)^2 t(2)^2 t(3)^2-2 t(1) t(2)^2 t(3)^2+t(2)^2 t(3)^2-t(1) t(3)^2-2 t(1)^2 t(2) t(3)^2+4 t(1) t(2) t(3)^2-t(2) t(3)^2-t(1)^2 t(3)+t(1) t(2)^2 t(3)-t(2)^2 t(3)+2 t(1) t(3)+t(1)^2 t(2) t(3)-4 t(1) t(2) t(3)+2 t(2) t(3)-t(3)-t(1)+t(1) t(2)-t(2)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial  q^{-13} -3 q^{-12} +6 q^{-11} -8 q^{-10} +11 q^{-9} -10 q^{-8} +10 q^{-7} -7 q^{-6} +5 q^{-5} -2 q^{-4} + q^{-3} (db)
Signature -6 (db)
HOMFLY-PT polynomial z^2 a^{12}+a^{12} z^{-2} +2 a^{12}-3 z^4 a^{10}-10 z^2 a^{10}-2 a^{10} z^{-2} -9 a^{10}+2 z^6 a^8+9 z^4 a^8+13 z^2 a^8+a^8 z^{-2} +7 a^8+z^6 a^6+4 z^4 a^6+4 z^2 a^6 (db)
Kauffman polynomial a^{16} z^4-a^{16} z^2+3 a^{15} z^5-3 a^{15} z^3+5 a^{14} z^6-7 a^{14} z^4+5 a^{14} z^2-2 a^{14}+5 a^{13} z^7-6 a^{13} z^5+3 a^{13} z^3+3 a^{12} z^8+a^{12} z^6-8 a^{12} z^4+5 a^{12} z^2-a^{12} z^{-2} +3 a^{12}+a^{11} z^9+6 a^{11} z^7-17 a^{11} z^5+17 a^{11} z^3-9 a^{11} z+2 a^{11} z^{-1} +6 a^{10} z^8-16 a^{10} z^6+22 a^{10} z^4-23 a^{10} z^2-2 a^{10} z^{-2} +11 a^{10}+a^9 z^9+3 a^9 z^7-14 a^9 z^5+15 a^9 z^3-9 a^9 z+2 a^9 z^{-1} +3 a^8 z^8-11 a^8 z^6+18 a^8 z^4-18 a^8 z^2-a^8 z^{-2} +7 a^8+2 a^7 z^7-6 a^7 z^5+4 a^7 z^3+a^6 z^6-4 a^6 z^4+4 a^6 z^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 \
-5          11
-7         21-1
-9        3  3
-11       42  -2
-13      63   3
-15     44    0
-17    76     1
-19   36      3
-21  35       -2
-23 14        3
-25 2         -2
-271          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-7 i=-5
r=-10 {\mathbb Z}
r=-9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-8 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=-7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

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See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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