From Knot Atlas
Jump to: navigation, search






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

Visit L11a304 at Knotilus!

Link Presentations

[edit Notes on L11a304's Link Presentations]

Planar diagram presentation X10,1,11,2 X2,11,3,12 X12,3,13,4 X16,5,17,6 X22,20,9,19 X20,14,21,13 X14,22,15,21 X18,7,19,8 X8,9,1,10 X4,15,5,16 X6,17,7,18
Gauss code {1, -2, 3, -10, 4, -11, 8, -9}, {9, -1, 2, -3, 6, -7, 10, -4, 11, -8, 5, -6, 7, -5}
A Braid Representative
A Morse Link Presentation L11a304 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(u v+1) \left(u^2 v^4-2 u^2 v^3+2 u^2 v^2+u v^3-2 u v^2+u v+2 v^2-2 v+1\right)}{u^{3/2} v^{5/2}} (db)
Jones polynomial \sqrt{q}-\frac{2}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{6}{q^{5/2}}+\frac{7}{q^{7/2}}-\frac{8}{q^{9/2}}+\frac{8}{q^{11/2}}-\frac{8}{q^{13/2}}+\frac{6}{q^{15/2}}-\frac{4}{q^{17/2}}+\frac{2}{q^{19/2}}-\frac{1}{q^{21/2}} (db)
Signature -5 (db)
HOMFLY-PT polynomial a^7 z^7+6 a^7 z^5+12 a^7 z^3+9 a^7 z+2 a^7 z^{-1} -a^5 z^9-8 a^5 z^7-24 a^5 z^5-33 a^5 z^3-19 a^5 z-3 a^5 z^{-1} +a^3 z^7+6 a^3 z^5+11 a^3 z^3+6 a^3 z+a^3 z^{-1} (db)
Kauffman polynomial a^{13} z^3-a^{13} z+2 a^{12} z^4-a^{12} z^2+3 a^{11} z^5-2 a^{11} z^3+a^{11} z+4 a^{10} z^6-5 a^{10} z^4+2 a^{10} z^2+5 a^9 z^7-12 a^9 z^5+10 a^9 z^3-3 a^9 z+4 a^8 z^8-9 a^8 z^6+2 a^8 z^4+a^8 z^2+3 a^7 z^9-9 a^7 z^7+9 a^7 z^5-13 a^7 z^3+10 a^7 z-2 a^7 z^{-1} +a^6 z^{10}+a^6 z^8-16 a^6 z^6+23 a^6 z^4-13 a^6 z^2+3 a^6+5 a^5 z^9-26 a^5 z^7+48 a^5 z^5-45 a^5 z^3+21 a^5 z-3 a^5 z^{-1} +a^4 z^{10}-2 a^4 z^8-9 a^4 z^6+25 a^4 z^4-17 a^4 z^2+3 a^4+2 a^3 z^9-12 a^3 z^7+24 a^3 z^5-19 a^3 z^3+6 a^3 z-a^3 z^{-1} +a^2 z^8-6 a^2 z^6+11 a^2 z^4-6 a^2 z^2+a^2 (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 \
2           1-1
0          1 1
-2         21 -1
-4        41  3
-6       43   -1
-8      43    1
-10     44     0
-12    44      0
-14   24       2
-16  24        -2
-18  2         2
-2012          -1
-221           1
Integral Khovanov Homology

(db, data source)

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