From Knot Atlas
Jump to: navigation, search






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

Visit L11a332 at Knotilus!

Link Presentations

[edit Notes on L11a332's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X20,5,21,6 X22,13,9,14 X18,21,19,22 X16,7,17,8 X14,17,15,18 X8,9,1,10 X6,15,7,16 X4,19,5,20
Gauss code {1, -2, 3, -11, 4, -10, 7, -9}, {9, -1, 2, -3, 5, -8, 10, -7, 8, -6, 11, -4, 6, -5}
A Braid Representative
A Morse Link Presentation L11a332 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{3 t(2)^3 t(1)^3-3 t(2)^2 t(1)^3+t(2) t(1)^3-3 t(2)^3 t(1)^2+8 t(2)^2 t(1)^2-5 t(2) t(1)^2+t(1)^2+t(2)^3 t(1)-5 t(2)^2 t(1)+8 t(2) t(1)-3 t(1)+t(2)^2-3 t(2)+3}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial \frac{2}{q^{9/2}}-\frac{1}{q^{7/2}}+\frac{1}{q^{29/2}}-\frac{4}{q^{27/2}}+\frac{7}{q^{25/2}}-\frac{11}{q^{23/2}}+\frac{14}{q^{21/2}}-\frac{15}{q^{19/2}}+\frac{15}{q^{17/2}}-\frac{12}{q^{15/2}}+\frac{8}{q^{13/2}}-\frac{6}{q^{11/2}} (db)
Signature -7 (db)
HOMFLY-PT polynomial a^{13} \left(-z^3\right)-2 a^{13} z+3 a^{11} z^5+11 a^{11} z^3+9 a^{11} z-2 a^9 z^7-10 a^9 z^5-15 a^9 z^3-6 a^9 z+a^9 z^{-1} -a^7 z^7-5 a^7 z^5-8 a^7 z^3-5 a^7 z-a^7 z^{-1} (db)
Kauffman polynomial a^{18} z^4+4 a^{17} z^5-3 a^{17} z^3+7 a^{16} z^6-7 a^{16} z^4+a^{16} z^2+8 a^{15} z^7-8 a^{15} z^5-a^{15} z^3+2 a^{15} z+7 a^{14} z^8-8 a^{14} z^6+a^{14} z^4+4 a^{13} z^9-12 a^{13} z^5+7 a^{13} z^3+a^{13} z+a^{12} z^{10}+10 a^{12} z^8-37 a^{12} z^6+42 a^{12} z^4-18 a^{12} z^2+7 a^{11} z^9-20 a^{11} z^7+18 a^{11} z^5-12 a^{11} z^3+7 a^{11} z+a^{10} z^{10}+5 a^{10} z^8-29 a^{10} z^6+38 a^{10} z^4-16 a^{10} z^2+3 a^9 z^9-11 a^9 z^7+13 a^9 z^5-9 a^9 z^3+3 a^9 z+a^9 z^{-1} +2 a^8 z^8-7 a^8 z^6+5 a^8 z^4+a^8 z^2-a^8+a^7 z^7-5 a^7 z^5+8 a^7 z^3-5 a^7 z+a^7 z^{-1} (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 \
-6           11
-8          21-1
-10         4  4
-12        42  -2
-14       84   4
-16      74    -3
-18     88     0
-20    67      1
-22   58       -3
-24  37        4
-26 14         -3
-28 3          3
-301           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-8 i=-6
r=-11 {\mathbb Z}
r=-10 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-9 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-8 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=-7 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-5 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-1 {\mathbb Z}_2^{2} {\mathbb Z}^{2}
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

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.