From Knot Atlas
Jump to: navigation, search






(Knotscape image)
See the full Thistlethwaite Link Table (up to 11 crossings).

Visit L11n297 at Knotilus!

Link Presentations

[edit Notes on L11n297's Link Presentations]

Planar diagram presentation X6172 X12,4,13,3 X15,20,16,21 X14,8,15,7 X21,10,22,5 X18,11,19,12 X9,17,10,16 X22,17,11,18 X8,19,9,20 X2536 X4,14,1,13
Gauss code {1, -10, 2, -11}, {10, -1, 4, -9, -7, 5}, {6, -2, 11, -4, -3, 7, 8, -6, 9, 3, -5, -8}
A Braid Representative
A Morse Link Presentation L11n297 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1) t(3)^2 t(2)^2-t(3)^2 t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-t(2)^2-t(1) t(3)^2 t(2)+2 t(1) t(3) t(2)-2 t(3) t(2)+t(2)+t(1) t(3)^2+t(1)-2 t(1) t(3)+t(3)-1}{\sqrt{t(1)} t(2) t(3)} (db)
Jones polynomial q^2-2 q+5-4 q^{-1} +7 q^{-2} -5 q^{-3} +5 q^{-4} -4 q^{-5} +2 q^{-6} - q^{-7} (db)
Signature 0 (db)
HOMFLY-PT polynomial -z^2 a^6-a^6 z^{-2} -2 a^6+2 z^4 a^4+7 z^2 a^4+4 a^4 z^{-2} +8 a^4-z^6 a^2-5 z^4 a^2-10 z^2 a^2-5 a^2 z^{-2} -10 a^2+z^4+3 z^2+2 z^{-2} +4 (db)
Kauffman polynomial a^5 z^9+a^3 z^9+2 a^6 z^8+5 a^4 z^8+3 a^2 z^8+a^7 z^7+a^3 z^7+2 a z^7-9 a^6 z^6-22 a^4 z^6-13 a^2 z^6-5 a^7 z^5-16 a^5 z^5-17 a^3 z^5-6 a z^5+11 a^6 z^4+30 a^4 z^4+24 a^2 z^4+5 z^4+8 a^7 z^3+29 a^5 z^3+29 a^3 z^3+10 a z^3+2 z^3 a^{-1} -5 a^6 z^2-22 a^4 z^2-27 a^2 z^2+z^2 a^{-2} -9 z^2-5 a^7 z-18 a^5 z-24 a^3 z-11 a z+3 a^6+12 a^4+15 a^2+7+a^7 z^{-1} +5 a^5 z^{-1} +9 a^3 z^{-1} +5 a z^{-1} -a^6 z^{-2} -4 a^4 z^{-2} -5 a^2 z^{-2} -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 \
5         11
3        21-1
1       3  3
-1      34  1
-3     41   3
-5    24    2
-7   33     0
-9  12      1
-11 13       -2
-13 1        1
-151         -1
Integral Khovanov Homology

(db, data source)

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

Read me first: Modifying Knot Pages

See/edit the Link Page master template (intermediate).

See/edit the Link_Splice_Base (expert).

Back to the top.