From Knot Atlas
Jump to: navigation, search






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

Visit L11a541 at Knotilus!

Link Presentations

[edit Notes on L11a541's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X20,12,21,11 X22,13,19,14 X18,22,9,21 X12,17,13,18 X8,16,5,15 X14,8,15,7 X16,19,17,20 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 8, -7}, {9, -3, 5, -4}, {11, -2, 3, -6, 4, -8, 7, -9, 6, -5}
A Braid Representative
A Morse Link Presentation L11a541 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(3)-1) (t(4)-1)^2 (-t(2) t(1)+t(2) t(4) t(1)-t(4) t(1)+2 t(1)+t(2)-2 t(2) t(4)+t(4)-1)}{\sqrt{t(1)} \sqrt{t(2)} \sqrt{t(3)} t(4)^{3/2}} (db)
Jones polynomial -q^{7/2}+5 q^{5/2}-13 q^{3/2}+18 \sqrt{q}-\frac{25}{\sqrt{q}}+\frac{25}{q^{3/2}}-\frac{27}{q^{5/2}}+\frac{20}{q^{7/2}}-\frac{15}{q^{9/2}}+\frac{7}{q^{11/2}}-\frac{3}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial -z a^7-a^7 z^{-1} +3 z^3 a^5+6 z a^5+5 a^5 z^{-1} +a^5 z^{-3} -3 z^5 a^3-8 z^3 a^3-12 z a^3-10 a^3 z^{-1} -3 a^3 z^{-3} +z^7 a+3 z^5 a+6 z^3 a+9 z a+9 a z^{-1} +3 a z^{-3} -z^5 a^{-1} -z^3 a^{-1} -2 z a^{-1} -3 a^{-1} z^{-1} - a^{-1} z^{-3} (db)
Kauffman polynomial -2 a^4 z^{10}-2 a^2 z^{10}-4 a^5 z^9-13 a^3 z^9-9 a z^9-4 a^6 z^8-12 a^4 z^8-24 a^2 z^8-16 z^8-3 a^7 z^7-4 a^5 z^7+5 a^3 z^7-7 a z^7-13 z^7 a^{-1} -a^8 z^6+3 a^6 z^6+24 a^4 z^6+48 a^2 z^6-5 z^6 a^{-2} +23 z^6+8 a^7 z^5+28 a^5 z^5+44 a^3 z^5+43 a z^5+18 z^5 a^{-1} -z^5 a^{-3} +3 a^8 z^4+9 a^6 z^4+2 a^4 z^4-11 a^2 z^4+2 z^4 a^{-2} -5 z^4-9 a^7 z^3-43 a^5 z^3-68 a^3 z^3-43 a z^3-9 z^3 a^{-1} -3 a^8 z^2-14 a^6 z^2-30 a^4 z^2-24 a^2 z^2-5 z^2+6 a^7 z+31 a^5 z+48 a^3 z+30 a z+7 z a^{-1} +a^8+6 a^6+18 a^4+21 a^2+9-2 a^7 z^{-1} -11 a^5 z^{-1} -18 a^3 z^{-1} -14 a z^{-1} -5 a^{-1} z^{-1} -3 a^4 z^{-2} -6 a^2 z^{-2} -3 z^{-2} +a^5 z^{-3} +3 a^3 z^{-3} +3 a z^{-3} + a^{-1} z^{-3} (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 \
8           11
6          4 -4
4         91 8
2        94  -5
0       169   7
-2      1515    0
-4     1210     2
-6    815      7
-8   712       -5
-10  210        8
-12 15         -4
-14 2          2
-161           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-2 i=0
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=0 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{16}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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.