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

Visit L10a63 at Knotilus!

Link Presentations

[edit Notes on L10a63's Link Presentations]

Planar diagram presentation X8192 X10,4,11,3 X20,10,7,9 X2738 X4,20,5,19 X14,5,15,6 X16,11,17,12 X18,13,19,14 X12,17,13,18 X6,15,1,16
Gauss code {1, -4, 2, -5, 6, -10}, {4, -1, 3, -2, 7, -9, 8, -6, 10, -7, 9, -8, 5, -3}
A Braid Representative
A Morse Link Presentation L10a63 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{u^2 v^4-2 u^2 v^3+2 u^2 v^2-2 u^2 v+u^2-2 u v^4+5 u v^3-5 u v^2+5 u v-2 u+v^4-2 v^3+2 v^2-2 v+1}{u v^2} (db)
Jones polynomial \frac{11}{q^{9/2}}-\frac{12}{q^{7/2}}+\frac{10}{q^{5/2}}+q^{3/2}-\frac{9}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{3}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{9}{q^{11/2}}-3 \sqrt{q}+\frac{5}{\sqrt{q}} (db)
Signature -3 (db)
HOMFLY-PT polynomial -z^3 a^7-2 z a^7-a^7 z^{-1} +2 z^5 a^5+7 z^3 a^5+8 z a^5+3 a^5 z^{-1} -z^7 a^3-5 z^5 a^3-10 z^3 a^3-9 z a^3-2 a^3 z^{-1} +z^5 a+3 z^3 a+2 z a (db)
Kauffman polynomial -z^4 a^{10}+z^2 a^{10}-3 z^5 a^9+3 z^3 a^9-5 z^6 a^8+6 z^4 a^8-3 z^2 a^8+a^8-6 z^7 a^7+10 z^5 a^7-10 z^3 a^7+5 z a^7-a^7 z^{-1} -4 z^8 a^6+2 z^6 a^6+6 z^4 a^6-9 z^2 a^6+3 a^6-z^9 a^5-10 z^7 a^5+32 z^5 a^5-35 z^3 a^5+16 z a^5-3 a^5 z^{-1} -7 z^8 a^4+14 z^6 a^4-3 z^4 a^4-6 z^2 a^4+3 a^4-z^9 a^3-7 z^7 a^3+29 z^5 a^3-32 z^3 a^3+15 z a^3-2 a^3 z^{-1} -3 z^8 a^2+6 z^6 a^2+z^4 a^2-3 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      54   -1
-6     75    2
-8    56     1
-10   46      -2
-12  25       3
-14 14        -3
-16 2         2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\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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See/edit the Link Page master template (intermediate).

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