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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 L11n404's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X16,9,17,10 X8,15,9,16 X4,17,1,18 X11,22,12,19 X10,4,11,3 X5,21,6,20 X21,5,22,18 X19,12,20,13 X2,14,3,13
Gauss code {1, -11, 7, -5}, {-10, 8, -9, 6}, {-8, -1, 2, -4, 3, -7, -6, 10, 11, -2, 4, -3, 5, 9}
A Braid Representative
A Morse Link Presentation L11n404 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) 0 (db)
Jones polynomial q^2-q+2+2 q^{-2} + q^{-3} - q^{-4} + q^{-5} - q^{-6} + q^{-7} - q^{-8} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^6 \left(-z^4\right)-4 a^6 z^2-a^6 z^{-2} -4 a^6+a^4 z^6+7 a^4 z^4+16 a^4 z^2+4 a^4 z^{-2} +13 a^4-a^2 z^6-7 a^2 z^4-16 a^2 z^2-5 a^2 z^{-2} -14 a^2+z^4+4 z^2+2 z^{-2} +5 (db)
Kauffman polynomial z^5 a^9-4 z^3 a^9+2 z a^9+z^6 a^8-4 z^4 a^8+2 z^2 a^8+z^7 a^7-5 z^5 a^7+6 z^3 a^7-4 z a^7+a^7 z^{-1} +z^8 a^6-6 z^6 a^6+10 z^4 a^6-10 z^2 a^6-a^6 z^{-2} +7 a^6+3 z^7 a^5-21 z^5 a^5+40 z^3 a^5-25 z a^5+5 a^5 z^{-1} +3 z^8 a^4-22 z^6 a^4+50 z^4 a^4-49 z^2 a^4-4 a^4 z^{-2} +22 a^4+z^9 a^3-4 z^7 a^3-7 z^5 a^3+34 z^3 a^3-31 z a^3+9 a^3 z^{-1} +3 z^8 a^2-22 z^6 a^2+52 z^4 a^2-53 z^2 a^2-5 a^2 z^{-2} +23 a^2+z^9 a-6 z^7 a+8 z^5 a+4 z^3 a-12 z a+5 a z^{-1} +z^8-7 z^6+16 z^4-16 z^2-2 z^{-2} +9 (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
3            0
1         21 1
-1       31   2
-3      152   2
-5     322    3
-7    121     0
-9   242      0
-11   11       0
-13 121        0
-15            0
-171           -1
Integral Khovanov Homology

(db, data source)

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