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

Visit L11n111 at Knotilus!

Link Presentations

[edit Notes on L11n111's Link Presentations]

Planar diagram presentation X6172 X3,12,4,13 X7,18,8,19 X19,22,20,5 X9,21,10,20 X21,9,22,8 X16,11,17,12 X14,17,15,18 X10,15,11,16 X2536 X13,4,14,1
Gauss code {1, -10, -2, 11}, {10, -1, -3, 6, -5, -9, 7, 2, -11, -8, 9, -7, 8, 3, -4, 5, -6, 4}
A Braid Representative
A Morse Link Presentation L11n111 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1) t(2)^5-2 t(1) t(2)^4-t(2)^4-t(1) t(2)^3+2 t(2)^3+2 t(1) t(2)^2-t(2)^2-t(1) t(2)-2 t(2)+1}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{1}{\sqrt{q}}+\frac{1}{q^{3/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{1}{q^{11/2}}+\frac{1}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{2}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -a^{11} z^{-1} +4 z a^9+4 a^9 z^{-1} -z^5 a^7-7 z^3 a^7-11 z a^7-6 a^7 z^{-1} +z^7 a^5+6 z^5 a^5+11 z^3 a^5+11 z a^5+5 a^5 z^{-1} -z^5 a^3-5 z^3 a^3-6 z a^3-2 a^3 z^{-1} (db)
Kauffman polynomial a^{12} z^6-4 a^{12} z^4+3 a^{12} z^2-a^{12}+2 a^{11} z^7-9 a^{11} z^5+9 a^{11} z^3-4 a^{11} z+a^{11} z^{-1} +a^{10} z^8-3 a^{10} z^6-4 a^{10} z^4+9 a^{10} z^2-3 a^{10}+3 a^9 z^7-18 a^9 z^5+29 a^9 z^3-18 a^9 z+4 a^9 z^{-1} +a^8 z^8-5 a^8 z^6+a^8 z^4+7 a^8 z^2-3 a^8+3 a^7 z^7-22 a^7 z^5+42 a^7 z^3-28 a^7 z+6 a^7 z^{-1} +a^6 z^8-6 a^6 z^6+6 a^6 z^4+a^6 z^2-a^6+3 a^5 z^7-19 a^5 z^5+33 a^5 z^3-22 a^5 z+5 a^5 z^{-1} +a^4 z^8-5 a^4 z^6+5 a^4 z^4-a^4+a^3 z^7-6 a^3 z^5+11 a^3 z^3-8 a^3 z+2 a^3 z^{-1} (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 \
0           11
-2            0
-4         21 1
-6       111  1
-8      111   -1
-10     221    1
-12    231     0
-14   111      1
-16  121       0
-18 11         0
-20 1          1
-221           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4 i=-2
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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\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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See/edit the Link Page master template (intermediate).

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