From Knot Atlas
Jump to: navigation, search






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

Visit L11a45 at Knotilus!

Link Presentations

[edit Notes on L11a45's Link Presentations]

Planar diagram presentation X6172 X14,7,15,8 X18,9,19,10 X20,15,21,16 X16,19,17,20 X8,17,9,18 X4,21,1,22 X12,6,13,5 X10,4,11,3 X22,12,5,11 X2,14,3,13
Gauss code {1, -11, 9, -7}, {8, -1, 2, -6, 3, -9, 10, -8, 11, -2, 4, -5, 6, -3, 5, -4, 7, -10}
A Braid Representative
A Morse Link Presentation L11a45 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{2 (t(1)-1) (t(2)-1) \left(t(2)^2-t(2)+1\right)^2}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -q^{7/2}+4 q^{5/2}-9 q^{3/2}+15 \sqrt{q}-\frac{21}{\sqrt{q}}+\frac{22}{q^{3/2}}-\frac{24}{q^{5/2}}+\frac{20}{q^{7/2}}-\frac{14}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{1}{q^{15/2}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^3 z^7+a z^7-a^5 z^5+3 a^3 z^5+3 a z^5-z^5 a^{-1} -2 a^5 z^3+3 a^3 z^3+3 a z^3-2 z^3 a^{-1} -a^5 z+2 a^3 z-z a^{-1} -a^5 z^{-1} +3 a^3 z^{-1} -2 a z^{-1} (db)
Kauffman polynomial -2 a^4 z^{10}-2 a^2 z^{10}-6 a^5 z^9-12 a^3 z^9-6 a z^9-7 a^6 z^8-12 a^4 z^8-14 a^2 z^8-9 z^8-4 a^7 z^7+7 a^5 z^7+15 a^3 z^7-4 a z^7-8 z^7 a^{-1} -a^8 z^6+16 a^6 z^6+31 a^4 z^6+27 a^2 z^6-4 z^6 a^{-2} +9 z^6+9 a^7 z^5+2 a^5 z^5+20 a z^5+12 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-11 a^6 z^4-19 a^4 z^4-10 a^2 z^4+5 z^4 a^{-2} +z^4-5 a^7 z^3+a^3 z^3-13 a z^3-8 z^3 a^{-1} +z^3 a^{-3} -a^8 z^2+2 a^6 z^2+5 a^4 z^2+a^2 z^2-2 z^2 a^{-2} -3 z^2-2 a^5 z-3 a^3 z+a z+2 z a^{-1} -a^6-3 a^4-3 a^2+a^5 z^{-1} +3 a^3 z^{-1} +2 a 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           11
6          3 -3
4         61 5
2        93  -6
0       126   6
-2      1211    -1
-4     1210     2
-6    812      4
-8   612       -6
-10  38        5
-12 16         -5
-14 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
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.