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

Visit K11a226 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X18,5,19,6 X20,7,21,8 X16,10,17,9 X14,12,15,11 X2,13,3,14 X10,16,11,15 X22,17,1,18 X8,19,9,20 X6,21,7,22
Gauss code 1, -7, 2, -1, 3, -11, 4, -10, 5, -8, 6, -2, 7, -6, 8, -5, 9, -3, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 12 18 20 16 14 2 10 22 8 6
A Braid Representative
A Morse Link Presentation K11a226 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a226/ThurstonBennequinNumber
Hyperbolic Volume 11.5144
A-Polynomial See Data:K11a226/A-polynomial

[edit Notes for K11a226's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 2

[edit Notes for K11a226's four dimensional invariants]

Polynomial invariants

Alexander polynomial -4 t^2+18 t-27+18 t^{-1} -4 t^{-2}
Conway polynomial -4 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 71, -2 }
Jones polynomial q^3-3 q^2+5 q-7+10 q^{-1} -11 q^{-2} +11 q^{-3} -9 q^{-4} +7 q^{-5} -4 q^{-6} +2 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -a^8+2 z^2 a^6+a^6-z^4 a^4+z^2 a^4+a^4-2 z^4 a^2-2 z^2 a^2-a^2-z^4+1+z^2 a^{-2}
Kauffman polynomial (db, data sources) a^4 z^{10}+a^2 z^{10}+2 a^5 z^9+5 a^3 z^9+3 a z^9+2 a^6 z^8+2 a^2 z^8+4 z^8+2 a^7 z^7-5 a^5 z^7-19 a^3 z^7-9 a z^7+3 z^7 a^{-1} +2 a^8 z^6-2 a^6 z^6-5 a^4 z^6-16 a^2 z^6+z^6 a^{-2} -14 z^6+a^9 z^5-2 a^7 z^5+10 a^5 z^5+31 a^3 z^5+8 a z^5-10 z^5 a^{-1} -5 a^8 z^4-a^6 z^4+13 a^4 z^4+25 a^2 z^4-3 z^4 a^{-2} +13 z^4-3 a^9 z^3-3 a^7 z^3-8 a^5 z^3-19 a^3 z^3-5 a z^3+6 z^3 a^{-1} +3 a^8 z^2+2 a^6 z^2-7 a^4 z^2-12 a^2 z^2+z^2 a^{-2} -5 z^2+2 a^9 z+2 a^7 z+2 a^5 z+4 a^3 z+2 a z-a^8-a^6+a^4+a^2+1
The A2 invariant Data:K11a226/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a226/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a229,}

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{556}{3} \frac{140}{3} -320 -\frac{2608}{3} -\frac{256}{3} -264 \frac{256}{3} 800 \frac{4448}{3} \frac{1120}{3} \frac{63031}{15} -\frac{11924}{15} \frac{122884}{45} \frac{1337}{9} \frac{6871}{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=-2 is the signature of K11a226. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
7           11
5          2 -2
3         31 2
1        42  -2
-1       63   3
-3      65    -1
-5     55     0
-7    46      2
-9   35       -2
-11  14        3
-13 13         -2
-15 1          1
-171           -1
Integral Khovanov Homology

(db, data source)

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

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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