From Knot Atlas
Jump to: navigation, search






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

Visit L11a518 at Knotilus!

Link Presentations

[edit Notes on L11a518's Link Presentations]

Planar diagram presentation X8192 X14,4,15,3 X22,9,13,10 X20,11,21,12 X12,19,7,20 X10,21,11,22 X18,14,19,13 X16,6,17,5 X2738 X4,16,5,15 X6,18,1,17
Gauss code {1, -9, 2, -10, 8, -11}, {9, -1, 3, -6, 4, -5}, {7, -2, 10, -8, 11, -7, 5, -4, 6, -3}
A Braid Representative
A Morse Link Presentation L11a518 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) \frac{-2 t(1)^2 t(3)^3+t(1) t(3)^3+t(1)^2 t(2) t(3)^3-t(1) t(2) t(3)^3+t(1)^2 t(3)^2+t(1)^2 t(2)^2 t(3)^2-t(1) t(2)^2 t(3)^2-2 t(1) t(3)^2-2 t(1)^2 t(2) t(3)^2+2 t(1) t(2) t(3)^2-t(2) t(3)^2+t(3)^2-t(1)^2 t(2)^2 t(3)+2 t(1) t(2)^2 t(3)-t(2)^2 t(3)+t(1) t(3)+t(1)^2 t(2) t(3)-2 t(1) t(2) t(3)+2 t(2) t(3)-t(3)-t(1) t(2)^2+2 t(2)^2+t(1) t(2)-t(2)}{t(1) t(2) t(3)^{3/2}} (db)
Jones polynomial -q^6+3 q^5+ q^{-5} -5 q^4- q^{-4} +8 q^3+4 q^{-3} -9 q^2-5 q^{-2} +10 q+8 q^{-1} -9 (db)
Signature 2 (db)
HOMFLY-PT polynomial z^6 a^{-2} +z^6-2 a^2 z^4+3 z^4 a^{-2} -z^4 a^{-4} +3 z^4+a^4 z^2-7 a^2 z^2+2 z^2 a^{-2} -2 z^2 a^{-4} +2 z^2+3 a^4-6 a^2+ a^{-2} +2+a^4 z^{-2} -2 a^2 z^{-2} + z^{-2} (db)
Kauffman polynomial z^3 a^{-7} +3 z^4 a^{-6} -z^2 a^{-6} +5 z^5 a^{-5} -3 z^3 a^{-5} +a^4 z^8-7 a^4 z^6+7 z^6 a^{-4} +17 a^4 z^4-10 z^4 a^{-4} -17 a^4 z^2+4 z^2 a^{-4} -a^4 z^{-2} +7 a^4+a^3 z^9-4 a^3 z^7+7 z^7 a^{-3} +a^3 z^5-13 z^5 a^{-3} +10 a^3 z^3+5 z^3 a^{-3} -9 a^3 z+2 a^3 z^{-1} +a^2 z^{10}-3 a^2 z^8+5 z^8 a^{-2} -3 a^2 z^6-8 z^6 a^{-2} +17 a^2 z^4-5 z^4 a^{-2} -20 a^2 z^2+7 z^2 a^{-2} -2 a^2 z^{-2} +11 a^2-2 a^{-2} +4 a z^9+3 z^9 a^{-1} -16 a z^7-5 z^7 a^{-1} +15 a z^5-4 z^5 a^{-1} +4 a z^3+3 z^3 a^{-1} -9 a z+2 a z^{-1} +z^{10}+z^8-11 z^6+8 z^4-z^2- z^{-2} +3 (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 \
13           1-1
11          2 2
9         31 -2
7        52  3
5       54   -1
3      54    1
1     67     1
-1    23      -1
-3   36       3
-5  12        -1
-7  3         3
-911          0
-111           1
Integral Khovanov Homology

(db, data source)

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