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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 L11a440's Link Presentations]

Planar diagram presentation X6172 X14,3,15,4 X12,15,5,16 X8,17,9,18 X16,7,17,8 X18,9,19,10 X22,19,13,20 X20,12,21,11 X10,22,11,21 X2536 X4,13,1,14
Gauss code {1, -10, 2, -11}, {10, -1, 5, -4, 6, -9, 8, -3}, {11, -2, 3, -5, 4, -6, 7, -8, 9, -7}
A Braid Representative
A Morse Link Presentation L11a440 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-u v^3 w^2+u v^3 w-u v^2 w^3+4 u v^2 w^2-3 u v^2 w+u v^2+u v w^3-3 u v w^2+4 u v w-2 u v+u w^2-2 u w+u+v^3 \left(-w^3\right)+2 v^3 w^2-v^3 w+2 v^2 w^3-4 v^2 w^2+3 v^2 w-v^2-v w^3+3 v w^2-4 v w+v-w^2+w}{\sqrt{u} v^{3/2} w^{3/2}} (db)
Jones polynomial - q^{-11} +3 q^{-10} -6 q^{-9} +10 q^{-8} -13 q^{-7} +17 q^{-6} -15 q^{-5} +14 q^{-4} -10 q^{-3} +7 q^{-2} -3 q^{-1} +1 (db)
Signature -4 (db)
HOMFLY-PT polynomial -z^2 a^{10}-2 a^{10}+3 z^4 a^8+9 z^2 a^8+a^8 z^{-2} +6 a^8-2 z^6 a^6-8 z^4 a^6-12 z^2 a^6-2 a^6 z^{-2} -9 a^6-z^6 a^4-z^4 a^4+5 z^2 a^4+a^4 z^{-2} +5 a^4+z^4 a^2+2 z^2 a^2 (db)
Kauffman polynomial a^{13} z^5-2 a^{13} z^3+a^{13} z+3 a^{12} z^6-6 a^{12} z^4+3 a^{12} z^2-a^{12}+4 a^{11} z^7-5 a^{11} z^5-a^{11} z^3+a^{11} z+4 a^{10} z^8-3 a^{10} z^6-3 a^{10} z^4+2 a^{10} z^2+3 a^9 z^9-7 a^9 z^5+9 a^9 z^3-3 a^9 z+a^8 z^{10}+8 a^8 z^8-27 a^8 z^6+38 a^8 z^4-23 a^8 z^2-a^8 z^{-2} +9 a^8+7 a^7 z^9-15 a^7 z^7+9 a^7 z^5+7 a^7 z^3-8 a^7 z+2 a^7 z^{-1} +a^6 z^{10}+9 a^6 z^8-37 a^6 z^6+54 a^6 z^4-38 a^6 z^2-2 a^6 z^{-2} +13 a^6+4 a^5 z^9-8 a^5 z^7+2 a^5 z^5+3 a^5 z^3-5 a^5 z+2 a^5 z^{-1} +5 a^4 z^8-15 a^4 z^6+16 a^4 z^4-14 a^4 z^2-a^4 z^{-2} +6 a^4+3 a^3 z^7-8 a^3 z^5+4 a^3 z^3+a^2 z^6-3 a^2 z^4+2 a^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 \
1           11
-1          2 -2
-3         51 4
-5        63  -3
-7       84   4
-9      76    -1
-11     108     2
-13    610      4
-15   47       -3
-17  26        4
-19 14         -3
-21 2          2
-231           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-9 {\mathbb Z}
r=-8 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-7 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{10}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\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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See/edit the Link Page master template (intermediate).

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