From Knot Atlas
Jump to: navigation, search






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

Visit L11a275 at Knotilus!

Link Presentations

[edit Notes on L11a275's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X14,5,15,6 X6,9,7,10 X18,8,19,7 X22,16,9,15 X20,18,21,17 X16,22,17,21 X8,20,1,19 X4,13,5,14
Gauss code {1, -2, 3, -11, 4, -5, 6, -10}, {5, -1, 2, -3, 11, -4, 7, -9, 8, -6, 10, -8, 9, -7}
A Braid Representative
A Morse Link Presentation L11a275 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{2 t(2)^3 t(1)^3-2 t(2)^2 t(1)^3-2 t(2)^3 t(1)^2+5 t(2)^2 t(1)^2-2 t(2) t(1)^2-2 t(2)^2 t(1)+5 t(2) t(1)-2 t(1)-2 t(2)+2}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial q^{7/2}-2 q^{5/2}+3 q^{3/2}-5 \sqrt{q}+\frac{6}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{7}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{4}{q^{11/2}}+\frac{2}{q^{13/2}}-\frac{1}{q^{15/2}} (db)
Signature -3 (db)
HOMFLY-PT polynomial a^5 z^5+4 a^5 z^3+4 a^5 z+a^5 z^{-1} -a^3 z^7-5 a^3 z^5-8 a^3 z^3-6 a^3 z-a^3 z^{-1} -a z^7-5 a z^5+z^5 a^{-1} -7 a z^3+4 z^3 a^{-1} -3 a z+3 z a^{-1} (db)
Kauffman polynomial -a^2 z^{10}-z^{10}-2 a^3 z^9-4 a z^9-2 z^9 a^{-1} -3 a^4 z^8+a^2 z^8-z^8 a^{-2} +3 z^8-4 a^5 z^7+3 a^3 z^7+19 a z^7+12 z^7 a^{-1} -4 a^6 z^6+5 a^4 z^6+6 a^2 z^6+6 z^6 a^{-2} +3 z^6-3 a^7 z^5+8 a^5 z^5+7 a^3 z^5-27 a z^5-23 z^5 a^{-1} -2 a^8 z^4+6 a^6 z^4+3 a^4 z^4-6 a^2 z^4-11 z^4 a^{-2} -12 z^4-a^9 z^3+2 a^7 z^3-6 a^5 z^3-11 a^3 z^3+14 a z^3+16 z^3 a^{-1} +a^8 z^2-4 a^6 z^2-4 a^4 z^2+a^2 z^2+6 z^2 a^{-2} +6 z^2+a^9 z-a^7 z+4 a^5 z+7 a^3 z-3 a z-4 z a^{-1} +a^4-a^5 z^{-1} -a^3 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 \
8           1-1
6          1 1
4         21 -1
2        31  2
0       32   -1
-2      53    2
-4     34     1
-6    44      0
-8   23       1
-10  24        -2
-12 13         2
-14 1          -1
-161           1
Integral Khovanov Homology

(db, data source)

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