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

Visit L11n4 at Knotilus!

Link Presentations

[edit Notes on L11n4's Link Presentations]

Planar diagram presentation X6172 X16,7,17,8 X4,17,1,18 X9,14,10,15 X3849 X5,11,6,10 X11,19,12,18 X13,20,14,21 X19,5,20,22 X21,12,22,13 X15,2,16,3
Gauss code {1, 11, -5, -3}, {-6, -1, 2, 5, -4, 6, -7, 10, -8, 4, -11, -2, 3, 7, -9, 8, -10, 9}
A Braid Representative
A Morse Link Presentation L11n4 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(2)^5-4 t(1) t(2)^4+6 t(1) t(2)^3-2 t(2)^3-2 t(1) t(2)^2+6 t(2)^2-4 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial q^{3/2}-3 \sqrt{q}+\frac{4}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{8}{q^{5/2}}-\frac{9}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{6}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^9 z^{-1} -a^7 z^3-4 a^7 z-3 a^7 z^{-1} +2 a^5 z^5+8 a^5 z^3+9 a^5 z+4 a^5 z^{-1} -a^3 z^7-5 a^3 z^5-9 a^3 z^3-8 a^3 z-2 a^3 z^{-1} +a z^5+3 a z^3+a z (db)
Kauffman polynomial -3 z^3 a^9+4 z a^9-a^9 z^{-1} -z^6 a^8-2 z^4 a^8+4 z^2 a^8-a^8-3 z^7 a^7+9 z^5 a^7-19 z^3 a^7+15 z a^7-3 a^7 z^{-1} -3 z^8 a^6+9 z^6 a^6-16 z^4 a^6+11 z^2 a^6-3 a^6-z^9 a^5-4 z^7 a^5+23 z^5 a^5-39 z^3 a^5+24 z a^5-4 a^5 z^{-1} -6 z^8 a^4+20 z^6 a^4-22 z^4 a^4+9 z^2 a^4-2 a^4-z^9 a^3-4 z^7 a^3+25 z^5 a^3-33 z^3 a^3+16 z a^3-2 a^3 z^{-1} -3 z^8 a^2+9 z^6 a^2-5 z^4 a^2+z^2 a^2-a^2-3 z^7 a+11 z^5 a-10 z^3 a+3 z a-z^6+3 z^4-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 \
4         1-1
2        2 2
0       21 -1
-2      52  3
-4     43   -1
-6    54    1
-8   34     1
-10  35      -2
-12 24       2
-14 2        -2
-162         2
Integral Khovanov Homology

(db, data source)

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