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

Visit L10a18 at Knotilus!

Link Presentations

[edit Notes on L10a18's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,10,13,9 X18,13,19,14 X16,7,17,8 X8,17,9,18 X20,15,5,16 X14,19,15,20 X2536 X4,12,1,11
Gauss code {1, -9, 2, -10}, {9, -1, 5, -6, 3, -2, 10, -3, 4, -8, 7, -5, 6, -4, 8, -7}
A Braid Representative
A Morse Link Presentation L10a18 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u-1) (v-1)^3 \left(v^2+1\right)}{\sqrt{u} v^{5/2}} (db)
Jones polynomial \frac{10}{q^{9/2}}-\frac{11}{q^{7/2}}+\frac{9}{q^{5/2}}+q^{3/2}-\frac{9}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{8}{q^{11/2}}-3 \sqrt{q}+\frac{5}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^7 \left(-z^3\right)-3 a^7 z-2 a^7 z^{-1} +2 a^5 z^5+8 a^5 z^3+11 a^5 z+5 a^5 z^{-1} -a^3 z^7-5 a^3 z^5-10 a^3 z^3-10 a^3 z-3 a^3 z^{-1} +a z^5+3 a z^3+2 a z (db)
Kauffman polynomial -z^4 a^{10}+2 z^2 a^{10}-a^{10}-2 z^5 a^9+2 z^3 a^9-3 z^6 a^8+2 z^4 a^8-4 z^7 a^7+6 z^5 a^7-8 z^3 a^7+5 z a^7-2 a^7 z^{-1} -3 z^8 a^6+z^6 a^6+6 z^4 a^6-11 z^2 a^6+5 a^6-z^9 a^5-7 z^7 a^5+24 z^5 a^5-29 z^3 a^5+17 z a^5-5 a^5 z^{-1} -6 z^8 a^4+11 z^6 a^4+2 z^4 a^4-11 z^2 a^4+5 a^4-z^9 a^3-6 z^7 a^3+26 z^5 a^3-29 z^3 a^3+16 z a^3-3 a^3 z^{-1} -3 z^8 a^2+6 z^6 a^2+2 z^4 a^2-4 z^2 a^2-3 z^7 a+10 z^5 a-10 z^3 a+4 z a-z^6+3 z^4-2 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 \
4          1-1
2         2 2
0        31 -2
-2       62  4
-4      55   0
-6     64    2
-8    45     1
-10   46      -2
-12  14       3
-14 14        -3
-16 1         1
-181          -1
Integral Khovanov Homology

(db, data source)

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