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

Visit L11a422 at Knotilus!

Link Presentations

[edit Notes on L11a422's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X18,13,19,14 X22,17,11,18 X16,7,17,8 X8,22,9,21 X20,10,21,9 X10,15,5,16 X14,19,15,20 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 5, -6, 7, -8}, {11, -2, 3, -9, 8, -5, 4, -3, 9, -7, 6, -4}
A Braid Representative
A Morse Link Presentation L11a422 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-t(2)^2 t(3)^4+t(1) t(3)^4-t(1) t(2) t(3)^4+2 t(2) t(3)^4-t(3)^4-t(1) t(2)^2 t(3)^3+2 t(2)^2 t(3)^3-3 t(1) t(3)^3+5 t(1) t(2) t(3)^3-5 t(2) t(3)^3+2 t(3)^3+2 t(1) t(2)^2 t(3)^2-3 t(2)^2 t(3)^2+3 t(1) t(3)^2-6 t(1) t(2) t(3)^2+6 t(2) t(3)^2-2 t(3)^2-2 t(1) t(2)^2 t(3)+3 t(2)^2 t(3)-2 t(1) t(3)+5 t(1) t(2) t(3)-5 t(2) t(3)+t(3)+t(1) t(2)^2-t(2)^2+t(1)-2 t(1) t(2)+t(2)}{\sqrt{t(1)} t(2) t(3)^2} (db)
Jones polynomial 1-4 q^{-1} +10 q^{-2} -14 q^{-3} +21 q^{-4} -22 q^{-5} +23 q^{-6} -19 q^{-7} +14 q^{-8} -8 q^{-9} +3 q^{-10} - q^{-11} (db)
Signature -4 (db)
HOMFLY-PT polynomial -z^2 a^{10}-a^{10} z^{-2} -2 a^{10}+3 z^4 a^8+8 z^2 a^8+4 a^8 z^{-2} +9 a^8-2 z^6 a^6-7 z^4 a^6-13 z^2 a^6-5 a^6 z^{-2} -14 a^6-z^6 a^4+6 z^2 a^4+2 a^4 z^{-2} +7 a^4+z^4 a^2+z^2 a^2 (db)
Kauffman polynomial z^5 a^{13}-2 z^3 a^{13}+z a^{13}+3 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+6 z^7 a^{11}-9 z^5 a^{11}+7 z^3 a^{11}-4 z a^{11}+a^{11} z^{-1} +8 z^8 a^{10}-13 z^6 a^{10}+14 z^4 a^{10}-11 z^2 a^{10}-a^{10} z^{-2} +5 a^{10}+6 z^9 a^9-20 z^5 a^9+33 z^3 a^9-21 z a^9+5 a^9 z^{-1} +2 z^{10} a^8+16 z^8 a^8-51 z^6 a^8+65 z^4 a^8-44 z^2 a^8-4 a^8 z^{-2} +18 a^8+13 z^9 a^7-20 z^7 a^7-7 z^5 a^7+34 z^3 a^7-29 z a^7+9 a^7 z^{-1} +2 z^{10} a^6+16 z^8 a^6-56 z^6 a^6+67 z^4 a^6-47 z^2 a^6-5 a^6 z^{-2} +21 a^6+7 z^9 a^5-10 z^7 a^5-5 z^5 a^5+13 z^3 a^5-13 z a^5+5 a^5 z^{-1} +8 z^8 a^4-20 z^6 a^4+18 z^4 a^4-14 z^2 a^4-2 a^4 z^{-2} +9 a^4+4 z^7 a^3-8 z^5 a^3+3 z^3 a^3+z^6 a^2-2 z^4 a^2+z^2 a^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 \
1           11
-1          3 -3
-3         71 6
-5        84  -4
-7       136   7
-9      1110    -1
-11     1211     1
-13    812      4
-15   611       -5
-17  28        6
-19 16         -5
-21 2          2
-231           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-5 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=-3 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{13}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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