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

Visit L11a259 at Knotilus!

Link Presentations

[edit Notes on L11a259's Link Presentations]

Planar diagram presentation X10,1,11,2 X14,5,15,6 X12,3,13,4 X18,8,19,7 X20,15,21,16 X22,17,9,18 X16,21,17,22 X4,13,5,14 X6,20,7,19 X2,9,3,10 X8,11,1,12
Gauss code {1, -10, 3, -8, 2, -9, 4, -11}, {10, -1, 11, -3, 8, -2, 5, -7, 6, -4, 9, -5, 7, -6}
A Braid Representative
A Morse Link Presentation L11a259 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{t(1)^2 t(2)^5-t(1) t(2)^5+t(1)^3 t(2)^4-3 t(1)^2 t(2)^4+2 t(1) t(2)^4-t(2)^4-t(1)^3 t(2)^3+3 t(1)^2 t(2)^3-3 t(1) t(2)^3+t(2)^3+t(1)^3 t(2)^2-3 t(1)^2 t(2)^2+3 t(1) t(2)^2-t(2)^2-t(1)^3 t(2)+2 t(1)^2 t(2)-3 t(1) t(2)+t(2)-t(1)^2+t(1)}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial -\frac{9}{q^{9/2}}+\frac{7}{q^{7/2}}-\frac{5}{q^{5/2}}+\frac{2}{q^{3/2}}+\frac{1}{q^{23/2}}-\frac{2}{q^{21/2}}+\frac{4}{q^{19/2}}-\frac{7}{q^{17/2}}+\frac{9}{q^{15/2}}-\frac{11}{q^{13/2}}+\frac{10}{q^{11/2}}-\frac{1}{\sqrt{q}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z^5 a^9-4 z^3 a^9-4 z a^9-a^9 z^{-1} +z^7 a^7+5 z^5 a^7+9 z^3 a^7+8 z a^7+3 a^7 z^{-1} +z^7 a^5+4 z^5 a^5+3 z^3 a^5-2 z a^5-2 a^5 z^{-1} -z^5 a^3-4 z^3 a^3-4 z a^3 (db)
Kauffman polynomial a^{14} z^4-2 a^{14} z^2+2 a^{13} z^5-3 a^{13} z^3+3 a^{12} z^6-5 a^{12} z^4+3 a^{12} z^2+4 a^{11} z^7-10 a^{11} z^5+13 a^{11} z^3-4 a^{11} z+4 a^{10} z^8-11 a^{10} z^6+14 a^{10} z^4-4 a^{10} z^2+a^{10}+3 a^9 z^9-8 a^9 z^7+9 a^9 z^5-3 a^9 z^3+a^9 z-a^9 z^{-1} +a^8 z^{10}+2 a^8 z^8-14 a^8 z^6+20 a^8 z^4-14 a^8 z^2+3 a^8+5 a^7 z^9-19 a^7 z^7+29 a^7 z^5-30 a^7 z^3+15 a^7 z-3 a^7 z^{-1} +a^6 z^{10}-8 a^6 z^6+8 a^6 z^4-6 a^6 z^2+3 a^6+2 a^5 z^9-6 a^5 z^7+3 a^5 z^5-3 a^5 z^3+6 a^5 z-2 a^5 z^{-1} +2 a^4 z^8-8 a^4 z^6+8 a^4 z^4-a^4 z^2+a^3 z^7-5 a^3 z^5+8 a^3 z^3-4 a^3 z (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          1 -1
-4         41 3
-6        42  -2
-8       53   2
-10      54    -1
-12     65     1
-14    46      2
-16   35       -2
-18  14        3
-20 13         -2
-22 1          1
-241           -1
Integral Khovanov Homology

(db, data source)

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