K11a322

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K11a321.gif

K11a321

K11a323.gif

K11a323

Contents

K11a322.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a322 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X16,5,17,6 X22,8,1,7 X20,9,21,10 X4,12,5,11 X18,13,19,14 X8,15,9,16 X2,17,3,18 X10,19,11,20 X14,21,15,22
Gauss code 1, -9, 2, -6, 3, -1, 4, -8, 5, -10, 6, -2, 7, -11, 8, -3, 9, -7, 10, -5, 11, -4
Dowker-Thistlethwaite code 6 12 16 22 20 4 18 8 2 10 14
A Braid Representative
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A Morse Link Presentation K11a322 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a322/ThurstonBennequinNumber
Hyperbolic Volume 17.6972
A-Polynomial See Data:K11a322/A-polynomial

[edit Notes for K11a322's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for K11a322's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-13 t^2+36 t-49+36 t^{-1} -13 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \left\{2,t^2+t+1\right\}
Determinant and Signature { 151, -2 }
Jones polynomial q^3-4 q^2+9 q-15+21 q^{-1} -24 q^{-2} +25 q^{-3} -21 q^{-4} +16 q^{-5} -10 q^{-6} +4 q^{-7} - q^{-8}
HOMFLY-PT polynomial (db, data sources) -z^4 a^6-z^2 a^6-2 a^6+z^6 a^4+2 z^4 a^4+5 z^2 a^4+4 a^4+z^6 a^2-2 z^2 a^2-2 a^2-2 z^4-z^2+1+z^2 a^{-2}
Kauffman polynomial (db, data sources) 3 a^4 z^{10}+3 a^2 z^{10}+9 a^5 z^9+16 a^3 z^9+7 a z^9+12 a^6 z^8+14 a^4 z^8+9 a^2 z^8+7 z^8+9 a^7 z^7-9 a^5 z^7-33 a^3 z^7-11 a z^7+4 z^7 a^{-1} +4 a^8 z^6-22 a^6 z^6-42 a^4 z^6-32 a^2 z^6+z^6 a^{-2} -15 z^6+a^9 z^5-14 a^7 z^5-3 a^5 z^5+24 a^3 z^5+3 a z^5-9 z^5 a^{-1} -4 a^8 z^4+18 a^6 z^4+40 a^4 z^4+29 a^2 z^4-2 z^4 a^{-2} +9 z^4-a^9 z^3+8 a^7 z^3+8 a^5 z^3-8 a^3 z^3-2 a z^3+5 z^3 a^{-1} -8 a^6 z^2-17 a^4 z^2-13 a^2 z^2+z^2 a^{-2} -3 z^2-3 a^7 z-3 a^5 z+a^3 z+a z+2 a^6+4 a^4+2 a^2+1
The A2 invariant Data:K11a322/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a322/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a147,}

Vassiliev invariants

V2 and V3: (2, -4)
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 -32 32 \frac{364}{3} \frac{68}{3} -256 -\frac{1664}{3} -\frac{32}{3} -192 \frac{256}{3} 512 \frac{2912}{3} \frac{544}{3} \frac{37351}{15} -\frac{9484}{15} \frac{77164}{45} \frac{761}{9} \frac{4231}{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 K11a322. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-101234χ
7           11
5          3 -3
3         61 5
1        93  -6
-1       126   6
-3      1310    -3
-5     1211     1
-7    913      4
-9   712       -5
-11  39        6
-13 17         -6
-15 3          3
-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}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-3 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-2 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=-1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{12}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a321.gif

K11a321

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K11a323