From Knot Atlas
Jump to: navigation, search






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

Visit L11a318 at Knotilus!

Link Presentations

[edit Notes on L11a318's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X8,9,1,10 X20,13,21,14 X16,8,17,7 X18,6,19,5 X6,18,7,17 X4,20,5,19 X22,15,9,16 X14,21,15,22
Gauss code {1, -2, 3, -9, 7, -8, 6, -4}, {4, -1, 2, -3, 5, -11, 10, -6, 8, -7, 9, -5, 11, -10}
A Braid Representative
A Morse Link Presentation L11a318 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(t(1)-1) (t(2)-1) \left(t(1) t(2)^4+t(1)^2 t(2)^3+t(1) t(2)^3+t(2)^3+t(1) t(2)^2+t(1)^2 t(2)+t(1) t(2)+t(2)+t(1)\right)}{t(1)^{3/2} t(2)^{5/2}} (db)
Jones polynomial -\frac{6}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{5}{q^{5/2}}-q^{3/2}+\frac{4}{q^{3/2}}+\frac{1}{q^{19/2}}-\frac{2}{q^{17/2}}+\frac{3}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{5}{q^{11/2}}+\sqrt{q}-\frac{3}{\sqrt{q}} (db)
Signature -5 (db)
HOMFLY-PT polynomial -z^5 a^7-4 z^3 a^7-3 z a^7+z^7 a^5+5 z^5 a^5+6 z^3 a^5-a^5 z^{-1} +z^7 a^3+6 z^5 a^3+12 z^3 a^3+10 z a^3+3 a^3 z^{-1} -z^5 a-5 z^3 a-7 z a-2 a z^{-1} (db)
Kauffman polynomial a^{12} z^2+2 a^{11} z^3+3 a^{10} z^4-2 a^{10} z^2+4 a^9 z^5-6 a^9 z^3+5 a^8 z^6-12 a^8 z^4+3 a^8 z^2+6 a^7 z^7-22 a^7 z^5+20 a^7 z^3-6 a^7 z+5 a^6 z^8-21 a^6 z^6+23 a^6 z^4-8 a^6 z^2+a^6+3 a^5 z^9-12 a^5 z^7+8 a^5 z^5+5 a^5 z^3-a^5 z^{-1} +a^4 z^{10}-a^4 z^8-16 a^4 z^6+37 a^4 z^4-21 a^4 z^2+3 a^4+4 a^3 z^9-26 a^3 z^7+57 a^3 z^5-51 a^3 z^3+19 a^3 z-3 a^3 z^{-1} +a^2 z^{10}-6 a^2 z^8+10 a^2 z^6-a^2 z^4-7 a^2 z^2+3 a^2+a z^9-8 a z^7+23 a z^5-28 a z^3+13 a z-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 \
4           11
2            0
0         31 2
-2        1   -1
-4       43   1
-6      33    0
-8     32     1
-10    23      1
-12   23       -1
-14  12        1
-16 12         -1
-18 1          1
-201           -1
Integral Khovanov Homology

(db, data source)

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