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

Visit L10a52 at Knotilus!

Link Presentations

[edit Notes on L10a52's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X20,10,7,9 X2738 X16,11,17,12 X12,5,13,6 X4,18,5,17 X14,20,15,19 X18,14,19,13 X6,15,1,16
Gauss code {1, -4, 2, -7, 6, -10}, {4, -1, 3, -2, 5, -6, 9, -8, 10, -5, 7, -9, 8, -3}
A Braid Representative
A Morse Link Presentation L10a52 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-3 t(1)^2 t(2)^3+6 t(1) t(2)^3-3 t(2)^3+4 t(1)^2 t(2)^2-7 t(1) t(2)^2+4 t(2)^2-3 t(1)^2 t(2)+6 t(1) t(2)-3 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2} (db)
Jones polynomial -4 q^{9/2}+\frac{1}{q^{9/2}}+8 q^{7/2}-\frac{4}{q^{7/2}}-12 q^{5/2}+\frac{7}{q^{5/2}}+15 q^{3/2}-\frac{12}{q^{3/2}}+q^{11/2}-16 \sqrt{q}+\frac{14}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^7 a^{-1} +2 a z^5-4 z^5 a^{-1} +z^5 a^{-3} -a^3 z^3+5 a z^3-6 z^3 a^{-1} +2 z^3 a^{-3} -a^3 z+2 a z-3 z a^{-1} +z a^{-3} +a^3 z^{-1} -a z^{-1} (db)
Kauffman polynomial -2 a z^9-2 z^9 a^{-1} -5 a^2 z^8-7 z^8 a^{-2} -12 z^8-4 a^3 z^7-9 a z^7-15 z^7 a^{-1} -10 z^7 a^{-3} -a^4 z^6+10 a^2 z^6+2 z^6 a^{-2} -8 z^6 a^{-4} +21 z^6+11 a^3 z^5+34 a z^5+40 z^5 a^{-1} +13 z^5 a^{-3} -4 z^5 a^{-5} +2 a^4 z^4-2 a^2 z^4+11 z^4 a^{-2} +8 z^4 a^{-4} -z^4 a^{-6} -2 z^4-9 a^3 z^3-27 a z^3-27 z^3 a^{-1} -7 z^3 a^{-3} +2 z^3 a^{-5} -a^4 z^2-2 a^2 z^2-6 z^2 a^{-2} -3 z^2 a^{-4} -4 z^2+a^3 z+5 a z+6 z a^{-1} +2 z a^{-3} -a^2+a^3 z^{-1} +a 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 \
12          1-1
10         3 3
8        51 -4
6       73  4
4      85   -3
2     87    1
0    79     2
-2   57      -2
-4  38       5
-6 14        -3
-8 3         3
-101          -1
Integral Khovanov Homology

(db, data source)

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