From Knot Atlas
Jump to: navigation, search






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

Visit L10a111 at Knotilus!

Link Presentations

[edit Notes on L10a111's Link Presentations]

Planar diagram presentation X12,1,13,2 X16,7,17,8 X8394 X2,18,3,17 X14,6,15,5 X6,12,7,11 X18,10,19,9 X20,15,11,16 X10,13,1,14 X4,19,5,20
Gauss code {1, -4, 3, -10, 5, -6, 2, -3, 7, -9}, {6, -1, 9, -5, 8, -2, 4, -7, 10, -8}
A Braid Representative
A Morse Link Presentation L10a111 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(2)^2 t(1)^2-2 t(2) t(1)^2+t(1)^2-2 t(2)^2 t(1)+3 t(2) t(1)-2 t(1)+t(2)^2-2 t(2)+1\right)}{t(1)^{3/2} t(2)^{3/2}} (db)
Jones polynomial -\frac{9}{q^{9/2}}-q^{7/2}+\frac{15}{q^{7/2}}+5 q^{5/2}-\frac{19}{q^{5/2}}-11 q^{3/2}+\frac{20}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{4}{q^{11/2}}+15 \sqrt{q}-\frac{20}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a z^7-2 a^3 z^5+3 a z^5-z^5 a^{-1} +a^5 z^3-4 a^3 z^3+3 a z^3-z^3 a^{-1} +a^5 z-2 a^3 z+a z+a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -4 a^3 z^9-4 a z^9-9 a^4 z^8-20 a^2 z^8-11 z^8-8 a^5 z^7-12 a^3 z^7-15 a z^7-11 z^7 a^{-1} -4 a^6 z^6+11 a^4 z^6+34 a^2 z^6-5 z^6 a^{-2} +14 z^6-a^7 z^5+12 a^5 z^5+34 a^3 z^5+39 a z^5+17 z^5 a^{-1} -z^5 a^{-3} +5 a^6 z^4-3 a^4 z^4-14 a^2 z^4+4 z^4 a^{-2} -2 z^4+a^7 z^3-8 a^5 z^3-22 a^3 z^3-19 a z^3-6 z^3 a^{-1} -2 a^6 z^2-a^4 z^2+2 a^2 z^2+z^2+2 a^5 z+4 a^3 z+2 a z+1-a z^{-1} - a^{-1} 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         4 -4
4        71 6
2       84  -4
0      127   5
-2     1010    0
-4    910     -1
-6   610      4
-8  39       -6
-10 16        5
-12 3         -3
-141          1
Integral Khovanov Homology

(db, data source)

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