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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 L11a504's Link Presentations]

Planar diagram presentation X8192 X2,9,3,10 X10,3,11,4 X14,5,15,6 X22,17,13,18 X20,15,21,16 X16,21,17,22 X12,19,7,20 X18,11,19,12 X6718 X4,13,5,14
Gauss code {1, -2, 3, -11, 4, -10}, {10, -1, 2, -3, 9, -8}, {11, -4, 6, -7, 5, -9, 8, -6, 7, -5}
A Braid Representative
A Morse Link Presentation L11a504 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 u^2 v^2 w^3-2 u^2 v^2 w^2+2 u^2 v^2 w-u^2 v^2-u^2 v w^3+u^2 v w^2-u^2 v w+u^2 v-u v^2 w^3+u v^2 w^2-u v^2 w+u v^2+2 u v w^3-2 u v w^2+2 u v w-2 u v-u w^3+u w^2-u w+u-v w^3+v w^2-v w+v+w^3-2 w^2+2 w-2}{u v w^{3/2}} (db)
Jones polynomial  q^{-4} - q^{-5} +5 q^{-6} -6 q^{-7} +10 q^{-8} -11 q^{-9} +12 q^{-10} -11 q^{-11} +9 q^{-12} -6 q^{-13} +3 q^{-14} - q^{-15} (db)
Signature -8 (db)
HOMFLY-PT polynomial -a^{14} z^{-2} -a^{14}-a^{12} z^6-3 a^{12} z^4+2 a^{12} z^2+4 a^{12} z^{-2} +9 a^{12}+a^{10} z^8+4 a^{10} z^6-a^{10} z^4-18 a^{10} z^2-5 a^{10} z^{-2} -19 a^{10}+a^8 z^8+7 a^8 z^6+18 a^8 z^4+21 a^8 z^2+2 a^8 z^{-2} +11 a^8 (db)
Kauffman polynomial z^3 a^{19}+3 z^4 a^{18}+6 z^5 a^{17}-4 z^3 a^{17}+z a^{17}+9 z^6 a^{16}-13 z^4 a^{16}+5 z^2 a^{16}-a^{16}+10 z^7 a^{15}-20 z^5 a^{15}+9 z^3 a^{15}-2 z a^{15}+a^{15} z^{-1} +8 z^8 a^{14}-17 z^6 a^{14}+3 z^4 a^{14}+3 z^2 a^{14}-a^{14} z^{-2} +4 z^9 a^{13}-3 z^7 a^{13}-23 z^5 a^{13}+33 z^3 a^{13}-19 z a^{13}+5 a^{13} z^{-1} +z^{10} a^{12}+6 z^8 a^{12}-32 z^6 a^{12}+37 z^4 a^{12}-20 z^2 a^{12}-4 a^{12} z^{-2} +13 a^{12}+5 z^9 a^{11}-16 z^7 a^{11}-z^5 a^{11}+39 z^3 a^{11}-35 z a^{11}+9 a^{11} z^{-1} +z^{10} a^{10}-z^8 a^{10}-13 z^6 a^{10}+36 z^4 a^{10}-39 z^2 a^{10}-5 a^{10} z^{-2} +22 a^{10}+z^9 a^9-3 z^7 a^9-4 z^5 a^9+20 z^3 a^9-19 z a^9+5 a^9 z^{-1} +z^8 a^8-7 z^6 a^8+18 z^4 a^8-21 z^2 a^8-2 a^8 z^{-2} +11 a^8 (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 \
-7           11
-9          110
-11         4  4
-13        21  -1
-15       84   4
-17      43    -1
-19     87     1
-21    56      1
-23   46       -2
-25  25        3
-27 14         -3
-29 2          2
-311           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-9 i=-7
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-8 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-7 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-6 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{8}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}_2 {\mathbb Z}
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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See/edit the Link Page master template (intermediate).

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