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

Visit L11n430 at Knotilus!

Link Presentations

[edit Notes on L11n430's Link Presentations]

Planar diagram presentation X8192 X9,20,10,21 X14,5,15,6 X12,14,7,13 X16,8,17,7 X22,18,13,17 X3,10,4,11 X18,11,19,12 X6,15,1,16 X4,20,5,19 X21,3,22,2
Gauss code {1, 11, -7, -10, 3, -9}, {5, -1, -2, 7, 8, -4}, {4, -3, 9, -5, 6, -8, 10, 2, -11, -6}
A Braid Representative
A Morse Link Presentation L11n430 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(v-1) (w-1) \left(u^2 w^2-u v w+u v-u w^2+u w-v\right)}{u v w^{3/2}} (db)
Jones polynomial -q^3+3 q^2-5 q+8-7 q^{-1} +9 q^{-2} -6 q^{-3} +5 q^{-4} -3 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial a^4 z^4+2 a^4 z^2+a^4 z^{-2} +a^4-a^2 z^6-4 a^2 z^4-6 a^2 z^2-z^2 a^{-2} -2 a^2 z^{-2} -4 a^2- a^{-2} +2 z^4+5 z^2+ z^{-2} +4 (db)
Kauffman polynomial a^6 z^6-3 a^6 z^4+a^6 z^2+3 a^5 z^7-10 a^5 z^5+6 a^5 z^3+4 a^4 z^8-15 a^4 z^6+16 a^4 z^4-9 a^4 z^2-a^4 z^{-2} +3 a^4+2 a^3 z^9-4 a^3 z^7-3 a^3 z^5+5 a^3 z^3+z^3 a^{-3} -2 a^3 z-z a^{-3} +2 a^3 z^{-1} +7 a^2 z^8-29 a^2 z^6+43 a^2 z^4+3 z^4 a^{-2} -27 a^2 z^2-4 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2+2 a^{-2} +2 a z^9-6 a z^7+z^7 a^{-1} +6 a z^5-z^5 a^{-1} +2 a z^3+4 z^3 a^{-1} -4 a z-3 z a^{-1} +2 a z^{-1} +3 z^8-13 z^6+27 z^4-21 z^2- z^{-2} +7 (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 \
7         1-1
5        2 2
3       31 -2
1      52  3
-1     45   1
-3    53    2
-5   36     3
-7  23      -1
-9 13       2
-11 2        -2
-131         1
Integral Khovanov Homology

(db, data source)

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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

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