From Knot Atlas
Jump to: navigation, search






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

Visit L11a477 at Knotilus!

Link Presentations

[edit Notes on L11a477's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X16,8,5,7 X18,9,19,10 X22,15,17,16 X20,13,21,14 X12,19,13,20 X14,21,15,22 X8,17,9,18 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {9, -4, 7, -6, 8, -5}, {10, -1, 3, -9, 4, -2, 11, -7, 6, -8, 5, -3}
A Braid Representative
A Morse Link Presentation L11a477 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{(u-1) (w-1) \left(v^2 w^3-2 v^2 w^2+2 v w^2-2 v w+2 w-1\right)}{\sqrt{u} v w^2} (db)
Jones polynomial  q^{-10} -2 q^{-9} +5 q^{-8} -8 q^{-7} +11 q^{-6} -12 q^{-5} +13 q^{-4} -10 q^{-3} +9 q^{-2} -q-5 q^{-1} +3 (db)
Signature -4 (db)
HOMFLY-PT polynomial a^8 z^4+4 a^8 z^2+a^8 z^{-2} +4 a^8-2 a^6 z^6-10 a^6 z^4-17 a^6 z^2-2 a^6 z^{-2} -12 a^6+a^4 z^8+6 a^4 z^6+14 a^4 z^4+17 a^4 z^2+a^4 z^{-2} +9 a^4-a^2 z^6-4 a^2 z^4-4 a^2 z^2-a^2 (db)
Kauffman polynomial a^{12} z^4-2 a^{12} z^2+a^{12}+2 a^{11} z^5-2 a^{11} z^3+3 a^{10} z^6-2 a^{10} z^4+4 a^9 z^7-5 a^9 z^5+4 a^9 z^3+4 a^8 z^8-6 a^8 z^6+6 a^8 z^4-6 a^8 z^2-a^8 z^{-2} +5 a^8+3 a^7 z^9-3 a^7 z^7-5 a^7 z^5+12 a^7 z^3-10 a^7 z+2 a^7 z^{-1} +a^6 z^{10}+6 a^6 z^8-29 a^6 z^6+43 a^6 z^4-35 a^6 z^2-2 a^6 z^{-2} +15 a^6+6 a^5 z^9-17 a^5 z^7+7 a^5 z^5+11 a^5 z^3-12 a^5 z+2 a^5 z^{-1} +a^4 z^{10}+5 a^4 z^8-33 a^4 z^6+51 a^4 z^4-36 a^4 z^2-a^4 z^{-2} +12 a^4+3 a^3 z^9-9 a^3 z^7+a^3 z^5+9 a^3 z^3-3 a^3 z+3 a^2 z^8-13 a^2 z^6+17 a^2 z^4-9 a^2 z^2+2 a^2+a z^7-4 a z^5+4 a z^3-a z (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 \
3           1-1
1          2 2
-1         31 -2
-3        62  4
-5       65   -1
-7      74    3
-9     56     1
-11    67      -1
-13   25       3
-15  36        -3
-17 14         3
-19 1          -1
-211           1
Integral Khovanov Homology

(db, data source)

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