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

Visit L11n387 at Knotilus!

Link Presentations

[edit Notes on L11n387's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X5,12,6,13 X8493 X11,19,12,22 X21,18,22,5 X9,20,10,21 X17,11,18,10 X19,17,20,16 X2,14,3,13
Gauss code {1, -11, 5, -3}, {-10, 8, -7, 6}, {-4, -1, 2, -5, -8, 9, -6, 4, 11, -2, 3, 10, -9, 7}
A Braid Representative
A Morse Link Presentation L11n387 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) (t(3)-1)^3}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial -q^3+4 q^2-6 q+11-10 q^{-1} +11 q^{-2} -9 q^{-3} +7 q^{-4} -4 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial -a^2 z^6+a^4 z^4-3 a^2 z^4+2 z^4+a^4 z^2-3 a^2 z^2-z^2 a^{-2} +3 z^2+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2} (db)
Kauffman polynomial 2 a^3 z^9+2 a z^9+5 a^4 z^8+9 a^2 z^8+4 z^8+4 a^5 z^7+3 a^3 z^7+a z^7+2 z^7 a^{-1} +a^6 z^6-13 a^4 z^6-27 a^2 z^6-13 z^6-11 a^5 z^5-23 a^3 z^5-16 a z^5-4 z^5 a^{-1} -2 a^6 z^4+5 a^4 z^4+25 a^2 z^4+4 z^4 a^{-2} +22 z^4+6 a^5 z^3+20 a^3 z^3+23 a z^3+10 z^3 a^{-1} +z^3 a^{-3} +a^6 z^2-a^4 z^2-9 a^2 z^2-4 z^2 a^{-2} -11 z^2-a^5 z-4 a^3 z-6 a z-4 z a^{-1} -z a^{-3} +1-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 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 \
7         1-1
5        3 3
3       31 -2
1      83  5
-1     67   1
-3    551   1
-5   46     2
-7  35      -2
-9 14       3
-11 3        -3
-131         1
Integral Khovanov Homology

(db, data source)

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