From Knot Atlas
Jump to: navigation, search






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

Visit L9a53 at Knotilus!

L9a53 is 9^3_{12} in the Rolfsen table of links.

Link Presentations

[edit Notes on L9a53's Link Presentations]

Planar diagram presentation X6172 X12,7,13,8 X4,13,1,14 X18,10,15,9 X8493 X16,5,17,6 X14,17,5,18 X10,16,11,15 X2,12,3,11
Gauss code {1, -9, 5, -3}, {8, -6, 7, -4}, {6, -1, 2, -5, 4, -8, 9, -2, 3, -7}
A Braid Representative
A Morse Link Presentation L9a53 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1) (t(3)-1)^3}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial q^4-4 q^3+8 q^2-9 q+12-10 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5} (db)
Signature 0 (db)
HOMFLY-PT polynomial -a^4 z^2-a^4+2 a^2 z^4+z^4 a^{-2} +4 a^2 z^2+z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +3 a^2+ a^{-2} -z^6-3 z^4-4 z^2-2 z^{-2} -3 (db)
Kauffman polynomial 2 a^2 z^8+2 z^8+4 a^3 z^7+11 a z^7+7 z^7 a^{-1} +3 a^4 z^6+7 a^2 z^6+8 z^6 a^{-2} +12 z^6+a^5 z^5-5 a^3 z^5-17 a z^5-7 z^5 a^{-1} +4 z^5 a^{-3} -6 a^4 z^4-22 a^2 z^4-11 z^4 a^{-2} +z^4 a^{-4} -28 z^4-2 a^5 z^3-a^3 z^3+3 a z^3-2 z^3 a^{-3} +5 a^4 z^2+17 a^2 z^2+5 z^2 a^{-2} +17 z^2+a^5 z+3 a^3 z+3 a z+z a^{-1} -2 a^4-6 a^2-2 a^{-2} -5-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + 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 \
9         11
7        3 -3
5       51 4
3      43  -1
1     85   3
-1    68    2
-3   44     0
-5  26      4
-7 14       -3
-9 2        2
-111         -1
Integral Khovanov Homology

(db, data source)

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

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.