From Knot Atlas
Jump to: navigation, search






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

Visit L11a325 at Knotilus!

Link Presentations

[edit Notes on L11a325's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X18,14,19,13 X14,8,15,7 X20,5,21,6 X4,19,5,20 X6,21,7,22 X16,9,17,10 X22,15,9,16 X8,18,1,17
Gauss code {1, -2, 3, -7, 6, -8, 5, -11}, {9, -1, 2, -3, 4, -5, 10, -9, 11, -4, 7, -6, 8, -10}
A Braid Representative
A Morse Link Presentation L11a325 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(1)^3 t(2)^5-t(1)^2 t(2)^5-2 t(1)^3 t(2)^4+4 t(1)^2 t(2)^4-2 t(1) t(2)^4+2 t(1)^3 t(2)^3-4 t(1)^2 t(2)^3+4 t(1) t(2)^3-t(2)^3-t(1)^3 t(2)^2+4 t(1)^2 t(2)^2-4 t(1) t(2)^2+2 t(2)^2-2 t(1)^2 t(2)+4 t(1) t(2)-2 t(2)-t(1)+1}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{5}{q^{3/2}}-\frac{9}{q^{5/2}}+\frac{10}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{13}{q^{11/2}}-\frac{11}{q^{13/2}}+\frac{9}{q^{15/2}}-\frac{6}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{1}{q^{21/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^7 z^7+5 a^7 z^5+8 a^7 z^3+5 a^7 z-a^5 z^9-7 a^5 z^7-18 a^5 z^5-21 a^5 z^3-9 a^5 z+a^5 z^{-1} +a^3 z^7+5 a^3 z^5+7 a^3 z^3+2 a^3 z-a^3 z^{-1} (db)
Kauffman polynomial -z^3 a^{13}-3 z^4 a^{12}-6 z^5 a^{11}+4 z^3 a^{11}-z a^{11}-9 z^6 a^{10}+13 z^4 a^{10}-5 z^2 a^{10}-10 z^7 a^9+19 z^5 a^9-7 z^3 a^9+z a^9-9 z^8 a^8+21 z^6 a^8-10 z^4 a^8+4 z^2 a^8-6 z^9 a^7+14 z^7 a^7-2 z^5 a^7-z^3 a^7-2 z a^7-2 z^{10} a^6-3 z^8 a^6+33 z^6 a^6-43 z^4 a^6+16 z^2 a^6-9 z^9 a^5+40 z^7 a^5-55 z^5 a^5+29 z^3 a^5-7 z a^5-a^5 z^{-1} -2 z^{10} a^4+5 z^8 a^4+8 z^6 a^4-25 z^4 a^4+11 z^2 a^4+a^4-3 z^9 a^3+16 z^7 a^3-28 z^5 a^3+18 z^3 a^3-3 z a^3-a^3 z^{-1} -z^8 a^2+5 z^6 a^2-8 z^4 a^2+4 z^2 a^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 \
2           1-1
0          2 2
-2         31 -2
-4        62  4
-6       54   -1
-8      85    3
-10     55     0
-12    68      -2
-14   46       2
-16  25        -3
-18 14         3
-20 2          -2
-221           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-6 i=-4
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\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.