K11a224

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K11a223.gif

K11a223

K11a225.gif

K11a225

Contents

K11a224.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a224 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X18,6,19,5 X20,8,21,7 X16,10,17,9 X2,12,3,11 X22,13,1,14 X10,16,11,15 X8,18,9,17 X6,20,7,19 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -9, 5, -8, 6, -2, 7, -11, 8, -5, 9, -3, 10, -4, 11, -7
Dowker-Thistlethwaite code 4 12 18 20 16 2 22 10 8 6 14
A Braid Representative
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A Morse Link Presentation K11a224 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a224/ThurstonBennequinNumber
Hyperbolic Volume 13.0759
A-Polynomial See Data:K11a224/A-polynomial

[edit Notes for K11a224's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -4

[edit Notes for K11a224's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+10 t^2-20 t+25-20 t^{-1} +10 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 89, 4 }
Jones polynomial -q^{11}+3 q^{10}-6 q^9+10 q^8-13 q^7+14 q^6-14 q^5+12 q^4-8 q^3+5 q^2-2 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -z^6 a^{-6} +z^4 a^{-2} -3 z^4 a^{-4} -2 z^4 a^{-6} +2 z^4 a^{-8} +3 z^2 a^{-2} -3 z^2 a^{-4} -z^2 a^{-6} +4 z^2 a^{-8} -z^2 a^{-10} +2 a^{-2} - a^{-4} - a^{-6} +2 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +2 z^9 a^{-5} +5 z^9 a^{-7} +3 z^9 a^{-9} +2 z^8 a^{-4} +z^8 a^{-6} +4 z^8 a^{-8} +5 z^8 a^{-10} +2 z^7 a^{-3} -3 z^7 a^{-5} -12 z^7 a^{-7} -2 z^7 a^{-9} +5 z^7 a^{-11} +z^6 a^{-2} -2 z^6 a^{-4} -z^6 a^{-6} -11 z^6 a^{-8} -10 z^6 a^{-10} +3 z^6 a^{-12} -6 z^5 a^{-3} +4 z^5 a^{-5} +20 z^5 a^{-7} -2 z^5 a^{-9} -11 z^5 a^{-11} +z^5 a^{-13} -4 z^4 a^{-2} -6 z^4 a^{-4} -z^4 a^{-6} +16 z^4 a^{-8} +9 z^4 a^{-10} -6 z^4 a^{-12} +4 z^3 a^{-3} -8 z^3 a^{-5} -17 z^3 a^{-7} +5 z^3 a^{-9} +8 z^3 a^{-11} -2 z^3 a^{-13} +5 z^2 a^{-2} +7 z^2 a^{-4} -3 z^2 a^{-6} -10 z^2 a^{-8} -3 z^2 a^{-10} +2 z^2 a^{-12} +z a^{-3} +3 z a^{-5} +3 z a^{-7} -z a^{-9} -2 z a^{-11} -2 a^{-2} - a^{-4} + a^{-6} +2 a^{-8} + a^{-10}
The A2 invariant Data:K11a224/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a224/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_92, K11a153, K11n35, K11n43,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {}

Vassiliev invariants

V2 and V3: (2, 5)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
8 40 32 \frac{604}{3} \frac{92}{3} 320 \frac{2992}{3} \frac{256}{3} 232 \frac{256}{3} 800 \frac{4832}{3} \frac{736}{3} \frac{75151}{15} -\frac{13124}{15} \frac{126724}{45} \frac{2321}{9} \frac{5791}{15}

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

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=4 is the signature of K11a224. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          2 2
19         41 -3
17        62  4
15       74   -3
13      76    1
11     77     0
9    57      -2
7   37       4
5  25        -3
3 14         3
1 1          -1
-11           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=9 {\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 Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11a223

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K11a225