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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 L11a387's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X18,11,19,12 X20,15,21,16 X22,17,9,18 X16,21,17,22 X12,19,13,20 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -9, 4, -3, 6, -8, 7, -5, 9, -6, 8, -7}
A Braid Representative
A Morse Link Presentation L11a387 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u v w^4-3 u v w^3+3 u v w^2-2 u v w+u v+u w^5-3 u w^4+4 u w^3-4 u w^2+3 u w-u+v w^5-3 v w^4+4 v w^3-4 v w^2+3 v w-v-w^5+2 w^4-3 w^3+3 w^2-2 w}{\sqrt{u} \sqrt{v} w^{5/2}} (db)
Jones polynomial  q^{-3} -3 q^{-4} +8 q^{-5} -11 q^{-6} +17 q^{-7} -16 q^{-8} +18 q^{-9} -15 q^{-10} +10 q^{-11} -6 q^{-12} +2 q^{-13} - q^{-14} (db)
Signature -6 (db)
HOMFLY-PT polynomial -2 a^{14} z^{-2} -a^{14}+4 a^{12} z^2+7 a^{12} z^{-2} +12 a^{12}-6 a^{10} z^4-22 a^{10} z^2-8 a^{10} z^{-2} -24 a^{10}+3 a^8 z^6+13 a^8 z^4+19 a^8 z^2+3 a^8 z^{-2} +13 a^8+a^6 z^6+3 a^6 z^4+2 a^6 z^2 (db)
Kauffman polynomial a^{17} z^5-3 a^{17} z^3+3 a^{17} z-a^{17} z^{-1} +2 a^{16} z^6-3 a^{16} z^4+a^{16}+3 a^{15} z^7-2 a^{15} z^5-3 a^{15} z^3+3 a^{15} z-a^{15} z^{-1} +4 a^{14} z^8-5 a^{14} z^6+7 a^{14} z^4-9 a^{14} z^2-2 a^{14} z^{-2} +7 a^{14}+3 a^{13} z^9+2 a^{13} z^7-14 a^{13} z^5+26 a^{13} z^3-21 a^{13} z+7 a^{13} z^{-1} +a^{12} z^{10}+10 a^{12} z^8-31 a^{12} z^6+45 a^{12} z^4-38 a^{12} z^2-7 a^{12} z^{-2} +22 a^{12}+7 a^{11} z^9-6 a^{11} z^7-23 a^{11} z^5+54 a^{11} z^3-45 a^{11} z+15 a^{11} z^{-1} +a^{10} z^{10}+12 a^{10} z^8-44 a^{10} z^6+64 a^{10} z^4-56 a^{10} z^2-8 a^{10} z^{-2} +28 a^{10}+4 a^9 z^9-2 a^9 z^7-19 a^9 z^5+31 a^9 z^3-24 a^9 z+8 a^9 z^{-1} +6 a^8 z^8-19 a^8 z^6+26 a^8 z^4-25 a^8 z^2-3 a^8 z^{-2} +13 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      89    1
-17     108     2
-19    58      3
-21   510       -5
-23  15        4
-25 15         -4
-27 1          1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{5}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-8 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-7 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-6 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\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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