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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 L11n354's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X11,18,12,19 X7,16,8,17 X15,8,16,9 X13,21,14,20 X19,22,20,15 X21,13,22,12 X17,14,18,5 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {-5, 4, -9, 3, -7, 6, -8, 7}, {10, -1, -4, 5, 11, -2, -3, 8, -6, 9}
A Braid Representative
A Morse Link Presentation L11n354 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(3)-1) \left(t(2) t(3)^3-t(3)^3+t(1) t(2)^2 t(3)^2-t(1) t(2) t(3)^2-t(2) t(3)^2-t(1) t(2) t(3)-t(2) t(3)+t(3)-t(1) t(2)^2+t(1) t(2)\right)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial - q^{-10} +2 q^{-9} -2 q^{-8} +3 q^{-7} - q^{-6} +2 q^{-5} + q^{-2} - q^{-1} +1 (db)
Signature -2 (db)
HOMFLY-PT polynomial -a^{10}+2 a^8 z^2+a^8 z^{-2} +2 a^8+a^6 z^2-2 a^6 z^{-2} -2 a^6-a^4 z^6-5 a^4 z^4-5 a^4 z^2+a^4 z^{-2} -a^4+a^2 z^4+4 a^2 z^2+2 a^2 (db)
Kauffman polynomial a^{11} z^7-5 a^{11} z^5+6 a^{11} z^3-2 a^{11} z+2 a^{10} z^8-11 a^{10} z^6+17 a^{10} z^4-11 a^{10} z^2+3 a^{10}+a^9 z^9-3 a^9 z^7-6 a^9 z^5+16 a^9 z^3-7 a^9 z+4 a^8 z^8-25 a^8 z^6+46 a^8 z^4-35 a^8 z^2-a^8 z^{-2} +12 a^8+a^7 z^9-4 a^7 z^7-4 a^7 z^5+19 a^7 z^3-14 a^7 z+2 a^7 z^{-1} +2 a^6 z^8-14 a^6 z^6+28 a^6 z^4-26 a^6 z^2-2 a^6 z^{-2} +12 a^6+a^5 z^7-8 a^5 z^5+14 a^5 z^3-10 a^5 z+2 a^5 z^{-1} +a^4 z^6-6 a^4 z^4+4 a^4 z^2-a^4 z^{-2} +2 a^4+a^3 z^7-5 a^3 z^5+5 a^3 z^3-a^3 z+a^2 z^6-5 a^2 z^4+6 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           11
-1            0
-3        121 0
-5       111  1
-7      231   0
-9     222    2
-11    241     1
-13   211      2
-15  131       1
-17 11         0
-19 1          1
-211           -1
Integral Khovanov Homology

(db, data source)

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