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 L11a28's page at Knotilus.

Visit L11a28's page at the original Knot Atlas.

Link Presentations

[edit Notes on L11a28's Link Presentations]

Planar diagram presentation X6172 X10,4,11,3 X14,8,15,7 X22,16,5,15 X16,9,17,10 X8,21,9,22 X20,17,21,18 X18,14,19,13 X12,20,13,19 X2536 X4,12,1,11
Gauss code {1, -10, 2, -11}, {10, -1, 3, -6, 5, -2, 11, -9, 8, -3, 4, -5, 7, -8, 9, -7, 6, -4}
A Braid Representative
A Morse Link Presentation L11a28 ML.gif

Polynomial invariants

Multivariable Alexander Polynomial (in u, v, w, ...) -\frac{(t(1)-1) (t(2)-1)^3 \left(t(2)^2-3 t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}} (db)
Jones polynomial -10 q^{9/2}+\frac{1}{q^{9/2}}+17 q^{7/2}-\frac{4}{q^{7/2}}-23 q^{5/2}+\frac{9}{q^{5/2}}+26 q^{3/2}-\frac{17}{q^{3/2}}-q^{13/2}+4 q^{11/2}-26 \sqrt{q}+\frac{22}{\sqrt{q}} (db)
Signature 1 (db)
HOMFLY-PT polynomial -z^3 a^{-5} -z a^{-5} - a^{-5} z^{-1} +2 z^5 a^{-3} -a^3 z^3+4 z^3 a^{-3} -a^3 z+5 z a^{-3} +3 a^{-3} z^{-1} -z^7 a^{-1} +2 a z^5-3 z^5 a^{-1} +4 a z^3-6 z^3 a^{-1} +4 a z-7 z a^{-1} +2 a z^{-1} -4 a^{-1} z^{-1} (db)
Kauffman polynomial z^5 a^{-7} -z^3 a^{-7} +4 z^6 a^{-6} -4 z^4 a^{-6} +z^2 a^{-6} +9 z^7 a^{-5} -13 z^5 a^{-5} +10 z^3 a^{-5} -4 z a^{-5} + a^{-5} z^{-1} +12 z^8 a^{-4} +a^4 z^6-18 z^6 a^{-4} -2 a^4 z^4+13 z^4 a^{-4} +a^4 z^2-5 z^2 a^{-4} + a^{-4} +8 z^9 a^{-3} +4 a^3 z^7+4 z^7 a^{-3} -9 a^3 z^5-32 z^5 a^{-3} +8 a^3 z^3+34 z^3 a^{-3} -3 a^3 z-16 z a^{-3} +3 a^{-3} z^{-1} +2 z^{10} a^{-2} +7 a^2 z^8+24 z^8 a^{-2} -13 a^2 z^6-58 z^6 a^{-2} +7 a^2 z^4+45 z^4 a^{-2} -a^2 z^2-17 z^2 a^{-2} +a^2+3 a^{-2} +6 a z^9+14 z^9 a^{-1} +a z^7-8 z^7 a^{-1} -27 a z^5-36 z^5 a^{-1} +30 a z^3+45 z^3 a^{-1} -13 a z-22 z a^{-1} +2 a z^{-1} +4 a^{-1} z^{-1} +2 z^{10}+19 z^8-50 z^6+37 z^4-13 z^2+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). 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 L11a28. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
14           11
12          3 -3
10         71 6
8        103  -7
6       137   6
4      1310    -3
2     1313     0
0    1115      4
-2   611       -5
-4  311        8
-6 16         -5
-8 3          3
-101           -1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=0 i=2
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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