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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 L11n288's Link Presentations]

Planar diagram presentation X6172 X5,12,6,13 X3849 X13,2,14,3 X14,7,15,8 X11,18,12,19 X9,21,10,20 X19,5,20,10 X4,15,1,16 X17,22,18,11 X21,16,22,17
Gauss code {1, 4, -3, -9}, {-2, -1, 5, 3, -7, 8}, {-6, 2, -4, -5, 9, 11, -10, 6, -8, 7, -11, 10}
A Braid Representative
A Morse Link Presentation L11n288 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(1) t(2)^2 t(3)^4+t(1) t(2) t(3)^4+t(1) t(2)^2 t(3)^3-t(3)^3-t(1) t(2)^2 t(3)^2+t(3)^2+t(1) t(2)^2 t(3)-t(3)-t(2)+1}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial - q^{-10} + q^{-9} -2 q^{-8} +3 q^{-7} -2 q^{-6} +4 q^{-5} -2 q^{-4} +3 q^{-3} - q^{-2} + q^{-1} (db)
Signature -6 (db)
HOMFLY-PT polynomial -z^2 a^{10}-a^{10} z^{-2} -2 a^{10}+z^6 a^8+6 z^4 a^8+11 z^2 a^8+4 a^8 z^{-2} +9 a^8-z^8 a^6-7 z^6 a^6-17 z^4 a^6-20 z^2 a^6-5 a^6 z^{-2} -14 a^6+z^6 a^4+6 z^4 a^4+11 z^2 a^4+2 a^4 z^{-2} +7 a^4 (db)
Kauffman polynomial a^{13} z+a^{12} z^2+2 a^{11} z^3-4 a^{11} z+a^{11} z^{-1} +a^{10} z^6-2 a^{10} z^4-4 a^{10} z^2-a^{10} z^{-2} +5 a^{10}+3 a^9 z^7-16 a^9 z^5+27 a^9 z^3-21 a^9 z+5 a^9 z^{-1} +3 a^8 z^8-17 a^8 z^6+32 a^8 z^4-32 a^8 z^2-4 a^8 z^{-2} +18 a^8+a^7 z^9-2 a^7 z^7-12 a^7 z^5+34 a^7 z^3-29 a^7 z+9 a^7 z^{-1} +4 a^6 z^8-25 a^6 z^6+51 a^6 z^4-45 a^6 z^2-5 a^6 z^{-2} +21 a^6+a^5 z^9-5 a^5 z^7+4 a^5 z^5+9 a^5 z^3-13 a^5 z+5 a^5 z^{-1} +a^4 z^8-7 a^4 z^6+17 a^4 z^4-18 a^4 z^2-2 a^4 z^{-2} +9 a^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 \
-1         11
-3          0
-5       31 2
-7     111  1
-9     42   2
-11   222    2
-13   32     1
-15 122      1
-17 12       -1
-1911        0
-211         -1
Integral Khovanov Homology

(db, data source)

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