Notice. The Knot Atlas is now recovering from a major crash. Hopefully all functionality will return slowly over the next few days. --Drorbn (talk) 21:23, 4 July 2013 (EDT)


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

Visit L11a30's page at Knotilus.

Visit L11a30's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a30's Link Presentations]

Planar diagram presentation X6172 X10,3,11,4 X16,8,17,7 X18,11,19,12 X22,19,5,20 X20,14,21,13 X12,22,13,21 X14,17,15,18 X8,16,9,15 X2536 X4,9,1,10
Gauss code {1, -10, 2, -11}, {10, -1, 3, -9, 11, -2, 4, -7, 6, -8, 9, -3, 8, -4, 5, -6, 7, -5}
A Braid Representative
A Morse Link Presentation L11a30 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{t(2)^5+2 t(1) t(2)^4-6 t(2)^4-9 t(1) t(2)^3+13 t(2)^3+13 t(1) t(2)^2-9 t(2)^2-6 t(1) t(2)+2 t(2)+t(1)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -\frac{14}{q^{9/2}}-q^{7/2}+\frac{17}{q^{7/2}}+3 q^{5/2}-\frac{20}{q^{5/2}}-7 q^{3/2}+\frac{20}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{4}{q^{13/2}}+\frac{8}{q^{11/2}}+12 \sqrt{q}-\frac{17}{\sqrt{q}} (db)
Signature -1 (db)
HOMFLY-PT polynomial a^7 (-z)+3 a^5 z^3+3 a^5 z+2 a^5 z^{-1} -2 a^3 z^5-4 a^3 z^3-7 a^3 z-4 a^3 z^{-1} -z a^{-3} -a z^5+2 a z^3+2 z^3 a^{-1} +4 a z+3 a z^{-1} - a^{-1} z^{-1} (db)
Kauffman polynomial -a^4 z^{10}-a^2 z^{10}-4 a^5 z^9-8 a^3 z^9-4 a z^9-6 a^6 z^8-15 a^4 z^8-15 a^2 z^8-6 z^8-4 a^7 z^7-4 a^5 z^7+a^3 z^7-4 a z^7-5 z^7 a^{-1} -a^8 z^6+12 a^6 z^6+42 a^4 z^6+40 a^2 z^6-3 z^6 a^{-2} +8 z^6+10 a^7 z^5+33 a^5 z^5+37 a^3 z^5+22 a z^5+7 z^5 a^{-1} -z^5 a^{-3} +2 a^8 z^4-4 a^6 z^4-35 a^4 z^4-41 a^2 z^4+5 z^4 a^{-2} -7 z^4-8 a^7 z^3-36 a^5 z^3-52 a^3 z^3-30 a z^3-4 z^3 a^{-1} +2 z^3 a^{-3} -a^8 z^2+a^6 z^2+12 a^4 z^2+17 a^2 z^2-2 z^2 a^{-2} +5 z^2+3 a^7 z+16 a^5 z+25 a^3 z+16 a z+3 z a^{-1} -z a^{-3} -a^6-2 a^4-3 a^2-1-2 a^5 z^{-1} -4 a^3 z^{-1} -3 a z^{-1} - a^{-1} 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). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=-1 is the signature of L11a30. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
8           11
6          2 -2
4         51 4
2        72  -5
0       105   5
-2      118    -3
-4     99     0
-6    811      3
-8   69       -3
-10  39        6
-12 15         -4
-14 3          3
-161           -1
Integral Khovanov Homology

(db, data source)

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

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