From Knot Atlas
Jump to: navigation, search






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

Visit L11a386 at Knotilus!

Link Presentations

[edit Notes on L11a386's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X14,7,15,8 X8,13,5,14 X18,12,19,11 X22,20,9,19 X20,16,21,15 X16,22,17,21 X12,18,13,17 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -4}, {11, -2, 5, -9, 4, -3, 7, -8, 9, -5, 6, -7, 8, -6}
A Braid Representative
A Morse Link Presentation L11a386 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{-3 t(1) t(3)^3+t(1) t(2) t(3)^3-3 t(2) t(3)^3+3 t(3)^3+7 t(1) t(3)^2-4 t(1) t(2) t(3)^2+7 t(2) t(3)^2-5 t(3)^2-7 t(1) t(3)+5 t(1) t(2) t(3)-7 t(2) t(3)+4 t(3)+3 t(1)-3 t(1) t(2)+3 t(2)-1}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}} (db)
Jones polynomial  q^{-6} -q^5-2 q^{-5} +4 q^4+7 q^{-4} -10 q^3-11 q^{-3} +15 q^2+19 q^{-2} -20 q-20 q^{-1} +22 (db)
Signature 0 (db)
HOMFLY-PT polynomial a^6 z^{-2} +a^6-3 a^4 z^2-a^4 z^{-2} -z^2 a^{-4} -4 a^4+3 a^2 z^4+2 z^4 a^{-2} +3 a^2 z^2-2 a^2 z^{-2} - a^{-2} z^{-2} -a^2-3 a^{-2} -z^6+4 z^2+3 z^{-2} +7 (db)
Kauffman polynomial a^2 z^{10}+z^{10}+3 a^3 z^9+8 a z^9+5 z^9 a^{-1} +3 a^4 z^8+12 a^2 z^8+10 z^8 a^{-2} +19 z^8+2 a^5 z^7+a^3 z^7+2 a z^7+12 z^7 a^{-1} +9 z^7 a^{-3} +a^6 z^6-3 a^4 z^6-30 a^2 z^6-13 z^6 a^{-2} +4 z^6 a^{-4} -43 z^6-4 a^5 z^5-12 a^3 z^5-39 a z^5-47 z^5 a^{-1} -15 z^5 a^{-3} +z^5 a^{-5} -4 a^6 z^4-4 a^4 z^4+26 a^2 z^4+4 z^4 a^{-2} -4 z^4 a^{-4} +34 z^4+16 a^3 z^3+53 a z^3+50 z^3 a^{-1} +12 z^3 a^{-3} -z^3 a^{-5} +6 a^6 z^2+6 a^4 z^2-11 a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} -14 z^2+4 a^5 z-10 a^3 z-34 a z-27 z a^{-1} -7 z a^{-3} -4 a^6-3 a^4+4 a^2+ a^{-2} +5-2 a^5 z^{-1} +2 a^3 z^{-1} +10 a z^{-1} +8 a^{-1} z^{-1} +2 a^{-3} z^{-1} +a^6 z^{-2} +a^4 z^{-2} -2 a^2 z^{-2} - a^{-2} z^{-2} -3 z^{-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 \
11           1-1
9          3 3
7         71 -6
5        83  5
3       127   -5
1      108    2
-1     1113     2
-3    89      -1
-5   311       8
-7  48        -4
-9 16         5
-11 1          -1
-131           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2 {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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.