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

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Link Presentations

[edit Notes on L11a497's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X14,8,15,7 X20,11,21,12 X18,10,5,9 X10,19,11,20 X8,16,9,15 X16,21,17,22 X22,17,19,18 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {6, -4, 8, -9}, {10, -1, 3, -7, 5, -6, 4, -2, 11, -3, 7, -8, 9, -5}
A Braid Representative
A Morse Link Presentation L11a497 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (w-1) \left(v w^4-2 v w^3+2 v w^2-2 v w+2 w^3-2 w^2+2 w-1\right)}{\sqrt{u} \sqrt{v} w^{5/2}} (db)
Jones polynomial  q^{-6} -q^5-3 q^{-5} +3 q^4+8 q^{-4} -7 q^3-11 q^{-3} +11 q^2+16 q^{-2} -15 q-17 q^{-1} +19 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^4 z^4+3 a^4 z^2+2 a^4 z^{-2} +4 a^4-2 a^2 z^6-z^6 a^{-2} -9 a^2 z^4-4 z^4 a^{-2} -16 a^2 z^2-6 z^2 a^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -14 a^2-4 a^{-2} +z^8+6 z^6+15 z^4+19 z^2+4 z^{-2} +14 (db)
Kauffman polynomial a^2 z^{10}+z^{10}+4 a^3 z^9+8 a z^9+4 z^9 a^{-1} +5 a^4 z^8+13 a^2 z^8+6 z^8 a^{-2} +14 z^8+3 a^5 z^7-3 a^3 z^7-10 a z^7+z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-12 a^4 z^6-49 a^2 z^6-11 z^6 a^{-2} +3 z^6 a^{-4} -50 z^6-7 a^5 z^5-14 a^3 z^5-19 a z^5-21 z^5 a^{-1} -8 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+10 a^4 z^4+68 a^2 z^4+12 z^4 a^{-2} -5 z^4 a^{-4} +72 z^4+3 a^5 z^3+22 a^3 z^3+50 a z^3+37 z^3 a^{-1} +4 z^3 a^{-3} -2 z^3 a^{-5} +3 a^6 z^2-13 a^4 z^2-50 a^2 z^2-11 z^2 a^{-2} +z^2 a^{-4} -46 z^2-17 a^3 z-35 a z-23 z a^{-1} -4 z a^{-3} +z a^{-5} -a^6+8 a^4+22 a^2+5 a^{-2} +19+5 a^3 z^{-1} +9 a z^{-1} +5 a^{-1} z^{-1} + a^{-3} z^{-1} -2 a^4 z^{-2} -5 a^2 z^{-2} - a^{-2} z^{-2} -4 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 \
11           1-1
9          2 2
7         51 -4
5        62  4
3       95   -4
1      106    4
-1     911     2
-3    78      -1
-5   49       5
-7  47        -3
-9 16         5
-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}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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