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

Visit L11a11 at Knotilus!

Link Presentations

[edit Notes on L11a11's Link Presentations]

Planar diagram presentation X6172 X20,7,21,8 X4,21,1,22 X14,10,15,9 X8493 X12,5,13,6 X22,13,5,14 X18,16,19,15 X16,11,17,12 X10,17,11,18 X2,20,3,19
Gauss code {1, -11, 5, -3}, {6, -1, 2, -5, 4, -10, 9, -6, 7, -4, 8, -9, 10, -8, 11, -2, 3, -7}
A Braid Representative
A Morse Link Presentation L11a11 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^4-5 t(2)^3+9 t(2)^2-5 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{16}{q^{9/2}}-q^{7/2}+\frac{23}{q^{7/2}}+5 q^{5/2}-\frac{27}{q^{5/2}}-12 q^{3/2}+\frac{27}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{9}{q^{11/2}}+18 \sqrt{q}-\frac{25}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)+3 a^5 z^3+3 a^5 z-3 a^3 z^5-6 a^3 z^3-4 a^3 z+a z^7+3 a z^5-z^5 a^{-1} +5 a z^3-z^3 a^{-1} +3 a z+a z^{-1} -z a^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial a^8 z^6-2 a^8 z^4+a^8 z^2+4 a^7 z^7-9 a^7 z^5+7 a^7 z^3-2 a^7 z+7 a^6 z^8-14 a^6 z^6+10 a^6 z^4-3 a^6 z^2+6 a^5 z^9-22 a^5 z^5+23 a^5 z^3-6 a^5 z+2 a^4 z^{10}+20 a^4 z^8-53 a^4 z^6+41 a^4 z^4-10 a^4 z^2+15 a^3 z^9-9 a^3 z^7-34 a^3 z^5+z^5 a^{-3} +35 a^3 z^3-8 a^3 z+2 a^2 z^{10}+28 a^2 z^8-64 a^2 z^6+5 z^6 a^{-2} +41 a^2 z^4-3 z^4 a^{-2} -9 a^2 z^2+9 a z^9+7 a z^7+12 z^7 a^{-1} -38 a z^5-16 z^5 a^{-1} +26 a z^3+7 z^3 a^{-1} -6 a z-2 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +15 z^8-21 z^6+9 z^4-3 z^2-1 (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 \
8           11
6          4 -4
4         81 7
2        104  -6
0       158   7
-2      1412    -2
-4     1313     0
-6    1014      4
-8   613       -7
-10  310        7
-12 16         -5
-14 3          3
-161           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-2 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{14} {\mathbb Z}^{14}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{15}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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