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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 L11a473's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X12,8,13,7 X22,15,17,16 X18,10,19,9 X8,18,9,17 X20,13,21,14 X14,21,15,22 X16,19,5,20 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {6, -5, 9, -7, 8, -4}, {10, -1, 3, -6, 5, -2, 11, -3, 7, -8, 4, -9}
A Braid Representative
A Morse Link Presentation L11a473 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (w-1) \left(v^2 w^3-v^2 w^2+v^2 w-v w^3+3 v w^2-3 v w+v-w^2+w-1\right)}{\sqrt{u} v w^2} (db)
Jones polynomial -q^5+3 q^4-7 q^3+12 q^2-15 q+19-17 q^{-1} +16 q^{-2} -11 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6} (db)
Signature 0 (db)
HOMFLY-PT polynomial z^8-2 a^2 z^6-z^6 a^{-2} +6 z^6+a^4 z^4-9 a^2 z^4-4 z^4 a^{-2} +15 z^4+3 a^4 z^2-15 a^2 z^2-6 z^2 a^{-2} +18 z^2+3 a^4-10 a^2-3 a^{-2} +10+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial a^2 z^{10}+z^{10}+3 a^3 z^9+8 a z^9+5 z^9 a^{-1} +4 a^4 z^8+10 a^2 z^8+8 z^8 a^{-2} +14 z^8+3 a^5 z^7+a^3 z^7-13 a z^7-5 z^7 a^{-1} +6 z^7 a^{-3} +a^6 z^6-7 a^4 z^6-32 a^2 z^6-22 z^6 a^{-2} +3 z^6 a^{-4} -49 z^6-8 a^5 z^5-15 a^3 z^5-7 z^5 a^{-1} -13 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+40 a^2 z^4+30 z^4 a^{-2} -5 z^4 a^{-4} +72 z^4+6 a^5 z^3+13 a^3 z^3+16 a z^3+21 z^3 a^{-1} +10 z^3 a^{-3} -2 z^3 a^{-5} +3 a^6 z^2-33 a^2 z^2-18 z^2 a^{-2} -48 z^2-a^5 z-7 a^3 z-13 a z-10 z a^{-1} -3 z a^{-3} -a^6+3 a^4+13 a^2+5 a^{-2} +15+2 a^3 z^{-1} +2 a z^{-1} -a^4 z^{-2} -2 a^2 z^{-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 \
11           1-1
9          2 2
7         51 -4
5        72  5
3       85   -3
1      117    4
-1     810     2
-3    89      -1
-5   510       5
-7  26        -4
-9 15         4
-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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
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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