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

Visit L10a4 at Knotilus!

Link Presentations

[edit Notes on L10a4's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X4,15,1,16 X12,6,13,5 X8493 X16,10,17,9 X20,18,5,17 X18,11,19,12 X10,19,11,20 X2,14,3,13
Gauss code {1, -10, 5, -3}, {4, -1, 2, -5, 6, -9, 8, -4, 10, -2, 3, -6, 7, -8, 9, -7}
A Braid Representative
A Morse Link Presentation L10a4 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1) \left(v^4-3 v^3+3 v^2-3 v+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial q^{11/2}-4 q^{9/2}+7 q^{7/2}-11 q^{5/2}+14 q^{3/2}-15 \sqrt{q}+\frac{13}{\sqrt{q}}-\frac{11}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{1}{q^{9/2}} (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^3-5 z^3 a^{-1} +2 z^3 a^{-3} -a^3 z+a z+a^3 z^{-1} -2 a z^{-1} +2 a^{-1} z^{-1} - a^{-3} z^{-1} (db)
Kauffman polynomial -2 a z^9-2 z^9 a^{-1} -5 a^2 z^8-6 z^8 a^{-2} -11 z^8-4 a^3 z^7-7 a z^7-11 z^7 a^{-1} -8 z^7 a^{-3} -a^4 z^6+11 a^2 z^6+2 z^6 a^{-2} -7 z^6 a^{-4} +21 z^6+11 a^3 z^5+29 a z^5+30 z^5 a^{-1} +8 z^5 a^{-3} -4 z^5 a^{-5} +2 a^4 z^4-3 a^2 z^4+9 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} -4 z^4-8 a^3 z^3-21 a z^3-17 z^3 a^{-1} -z^3 a^{-3} +3 z^3 a^{-5} -a^4 z^2-2 a^2 z^2-6 z^2 a^{-2} -2 z^2 a^{-4} -5 z^2+a^3 z-2 z a^{-1} -z a^{-3} +1+a^3 z^{-1} +2 a z^{-1} +2 a^{-1} z^{-1} + a^{-3} 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        41 -3
6       73  4
4      74   -3
2     87    1
0    79     2
-2   46      -2
-4  37       4
-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}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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See/edit the Link Page master template (intermediate).

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