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

Visit L10a76 at Knotilus!

Link Presentations

[edit Notes on L10a76's Link Presentations]

Planar diagram presentation X8192 X14,9,15,10 X6718 X20,15,7,16 X16,6,17,5 X4,20,5,19 X10,4,11,3 X12,17,13,18 X18,11,19,12 X2,14,3,13
Gauss code {1, -10, 7, -6, 5, -3}, {3, -1, 2, -7, 9, -8, 10, -2, 4, -5, 8, -9, 6, -4}
A Braid Representative
A Morse Link Presentation L10a76 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{u^2 v^4-4 u^2 v^3+5 u^2 v^2-2 u^2 v-u v^4+7 u v^3-11 u v^2+7 u v-u-2 v^3+5 v^2-4 v+1}{u v^2} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-9 q^{3/2}+12 \sqrt{q}-\frac{17}{\sqrt{q}}+\frac{17}{q^{3/2}}-\frac{16}{q^{5/2}}+\frac{13}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{4}{q^{11/2}}-\frac{1}{q^{13/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+4 a z^5-z^5 a^{-1} +a^5 z^3-5 a^3 z^3+7 a z^3-2 z^3 a^{-1} +a^5 z-4 a^3 z+6 a z-2 z a^{-1} -a^3 z^{-1} +3 a z^{-1} -2 a^{-1} z^{-1} (db)
Kauffman polynomial -3 a^3 z^9-3 a z^9-7 a^4 z^8-15 a^2 z^8-8 z^8-7 a^5 z^7-9 a^3 z^7-10 a z^7-8 z^7 a^{-1} -4 a^6 z^6+8 a^4 z^6+29 a^2 z^6-4 z^6 a^{-2} +13 z^6-a^7 z^5+11 a^5 z^5+28 a^3 z^5+32 a z^5+15 z^5 a^{-1} -z^5 a^{-3} +6 a^6 z^4+a^4 z^4-17 a^2 z^4+5 z^4 a^{-2} -7 z^4+a^7 z^3-5 a^5 z^3-23 a^3 z^3-28 a z^3-10 z^3 a^{-1} +z^3 a^{-3} -2 a^6 z^2-3 a^4 z^2-a^2 z^2+2 a^5 z+7 a^3 z+10 a z+5 z a^{-1} +a^4+3 a^2+3-a^3 z^{-1} -3 a z^{-1} -2 a^{-1} 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 \
8          11
6         3 -3
4        61 5
2       74  -3
0      105   5
-2     88    0
-4    89     -1
-6   58      3
-8  38       -5
-10 15        4
-12 3         -3
-141          1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\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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