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

Visit L11a503 at Knotilus!

Link Presentations

[edit Notes on L11a503's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X18,12,19,11 X12,18,7,17 X20,16,21,15 X22,20,13,19 X16,22,17,21 X6718 X4,13,5,14
Gauss code {1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 5, -6}, {11, -4, 7, -9, 6, -5, 8, -7, 9, -8}
A Braid Representative
A Morse Link Presentation L11a503 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^2 t(3)^3-t(1) t(3)^3-t(1)^2 t(2) t(3)^3+t(1) t(2) t(3)^3+t(3)^3-3 t(1)^2 t(2)^2 t(3)^2+2 t(1) t(2)^2 t(3)^2+2 t(1) t(3)^2+2 t(1)^2 t(2) t(3)^2-4 t(1) t(2) t(3)^2+2 t(2) t(3)^2-3 t(3)^2+3 t(1)^2 t(2)^2 t(3)-2 t(1) t(2)^2 t(3)-2 t(1) t(3)-2 t(1)^2 t(2) t(3)+4 t(1) t(2) t(3)-2 t(2) t(3)+3 t(3)-t(1)^2 t(2)^2+t(1) t(2)^2-t(1) t(2)+t(2)-1}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial -q^4+2 q^3-5 q^2+9 q-11+15 q^{-1} -14 q^{-2} +14 q^{-3} -10 q^{-4} +7 q^{-5} -3 q^{-6} + q^{-7} (db)
Signature -2 (db)
HOMFLY-PT polynomial a^4 z^6+4 a^4 z^4+6 a^4 z^2+2 a^4 z^{-2} +5 a^4-a^2 z^8-6 a^2 z^6-15 a^2 z^4-z^4 a^{-2} -21 a^2 z^2-4 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -16 a^2-4 a^{-2} +2 z^6+10 z^4+18 z^2+4 z^{-2} +15 (db)
Kauffman polynomial a^2 z^{10}+z^{10}+4 a^3 z^9+6 a z^9+2 z^9 a^{-1} +7 a^4 z^8+9 a^2 z^8+2 z^8 a^{-2} +4 z^8+7 a^5 z^7-a^3 z^7-12 a z^7-3 z^7 a^{-1} +z^7 a^{-3} +6 a^6 z^6-12 a^4 z^6-35 a^2 z^6-8 z^6 a^{-2} -25 z^6+3 a^7 z^5-8 a^5 z^5-15 a^3 z^5-11 a z^5-12 z^5 a^{-1} -5 z^5 a^{-3} +a^8 z^4-8 a^6 z^4+11 a^4 z^4+47 a^2 z^4+10 z^4 a^{-2} +37 z^4-2 a^7 z^3+2 a^5 z^3+23 a^3 z^3+39 a z^3+28 z^3 a^{-1} +8 z^3 a^{-3} -a^8 z^2+6 a^6 z^2-9 a^4 z^2-39 a^2 z^2-7 z^2 a^{-2} -30 z^2+a^5 z-16 a^3 z-33 a z-21 z a^{-1} -5 z a^{-3} -2 a^6+6 a^4+20 a^2+4 a^{-2} +17+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -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 \
9           1-1
7          1 1
5         41 -3
3        51  4
1       64   -2
-1      95    4
-3     78     1
-5    77      0
-7   48       4
-9  36        -3
-11  4         4
-1313          -2
-151           1
Integral Khovanov Homology

(db, data source)

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