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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 L11a413's Link Presentations]

Planar diagram presentation X6172 X12,3,13,4 X20,13,21,14 X22,19,11,20 X10,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X14,21,15,22 X2536 X4,11,1,12
Gauss code {1, -10, 2, -11}, {10, -1, 7, -6, 8, -5}, {11, -2, 3, -9, 5, -7, 6, -8, 4, -3, 9, -4}
A Braid Representative
A Morse Link Presentation L11a413 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u v^2 w^3-2 u v^2 w^2+u v^2 w+u v w^4-4 u v w^3+6 u v w^2-4 u v w+u v+u w^3-3 u w^2+3 u w-u+v^2 w^4-3 v^2 w^3+3 v^2 w^2-v^2 w-v w^4+4 v w^3-6 v w^2+4 v w-v-w^3+2 w^2-2 w}{\sqrt{u} v w^2} (db)
Jones polynomial - q^{-14} +3 q^{-13} -7 q^{-12} +12 q^{-11} -16 q^{-10} +19 q^{-9} -18 q^{-8} +17 q^{-7} -11 q^{-6} +8 q^{-5} -3 q^{-4} + q^{-3} (db)
Signature -6 (db)
HOMFLY-PT polynomial -a^{14} z^{-2} -a^{14}+4 a^{12} z^2+4 a^{12} z^{-2} +9 a^{12}-6 a^{10} z^4-20 a^{10} z^2-5 a^{10} z^{-2} -19 a^{10}+3 a^8 z^6+13 a^8 z^4+19 a^8 z^2+2 a^8 z^{-2} +11 a^8+a^6 z^6+3 a^6 z^4+2 a^6 z^2 (db)
Kauffman polynomial a^{17} z^5-2 a^{17} z^3+a^{17} z+3 a^{16} z^6-5 a^{16} z^4+3 a^{16} z^2-a^{16}+5 a^{15} z^7-7 a^{15} z^5+4 a^{15} z^3-2 a^{15} z+a^{15} z^{-1} +5 a^{14} z^8-2 a^{14} z^6-6 a^{14} z^4+5 a^{14} z^2-a^{14} z^{-2} +3 a^{13} z^9+8 a^{13} z^7-27 a^{13} z^5+31 a^{13} z^3-19 a^{13} z+5 a^{13} z^{-1} +a^{12} z^{10}+11 a^{12} z^8-24 a^{12} z^6+21 a^{12} z^4-16 a^{12} z^2-4 a^{12} z^{-2} +13 a^{12}+7 a^{11} z^9-2 a^{11} z^7-29 a^{11} z^5+50 a^{11} z^3-35 a^{11} z+9 a^{11} z^{-1} +a^{10} z^{10}+12 a^{10} z^8-39 a^{10} z^6+51 a^{10} z^4-43 a^{10} z^2-5 a^{10} z^{-2} +22 a^{10}+4 a^9 z^9-2 a^9 z^7-17 a^9 z^5+28 a^9 z^3-19 a^9 z+5 a^9 z^{-1} +6 a^8 z^8-19 a^8 z^6+26 a^8 z^4-23 a^8 z^2-2 a^8 z^{-2} +11 a^8+3 a^7 z^7-7 a^7 z^5+3 a^7 z^3+a^6 z^6-3 a^6 z^4+2 a^6 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 \
-5           11
-7          31-2
-9         5  5
-11        63  -3
-13       115   6
-15      87    -1
-17     1110     1
-19    710      3
-21   59       -4
-23  27        5
-25 15         -4
-27 2          2
-291           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-7 i=-5
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-8 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-7 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-6 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{11}
r=-5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z} {\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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