K11a303

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K11a302.gif

K11a302

K11a304.gif

K11a304

Contents

K11a303.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a303 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X18,5,19,6 X22,8,1,7 X16,10,17,9 X4,11,5,12 X8,14,9,13 X20,16,21,15 X12,18,13,17 X2,19,3,20 X14,22,15,21
Gauss code 1, -10, 2, -6, 3, -1, 4, -7, 5, -2, 6, -9, 7, -11, 8, -5, 9, -3, 10, -8, 11, -4
Dowker-Thistlethwaite code 6 10 18 22 16 4 8 20 12 2 14
A Braid Representative
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A Morse Link Presentation K11a303 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:K11a303/ThurstonBennequinNumber
Hyperbolic Volume 17.779
A-Polynomial See Data:K11a303/A-polynomial

[edit Notes for K11a303's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a303's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+15 t^2-34 t+45-34 t^{-1} +15 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-3 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 149, 0 }
Jones polynomial -q^7+4 q^6-9 q^5+15 q^4-20 q^3+24 q^2-24 q+21-16 q^{-1} +10 q^{-2} -4 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) -2 z^6 a^{-2} -z^6+a^2 z^4-6 z^4 a^{-2} +3 z^4 a^{-4} -z^4+a^2 z^2-7 z^2 a^{-2} +6 z^2 a^{-4} -z^2 a^{-6} +a^2-2 a^{-2} +3 a^{-4} - a^{-6}
Kauffman polynomial (db, data sources) 3 z^{10} a^{-2} +3 z^{10} a^{-4} +11 z^9 a^{-1} +17 z^9 a^{-3} +6 z^9 a^{-5} +20 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8 a^{-6} +17 z^8+16 a z^7-8 z^7 a^{-1} -41 z^7 a^{-3} -16 z^7 a^{-5} +z^7 a^{-7} +10 a^2 z^6-69 z^6 a^{-2} -45 z^6 a^{-4} -13 z^6 a^{-6} -27 z^6+4 a^3 z^5-22 a z^5-19 z^5 a^{-1} +18 z^5 a^{-3} +8 z^5 a^{-5} -3 z^5 a^{-7} +a^4 z^4-8 a^2 z^4+58 z^4 a^{-2} +52 z^4 a^{-4} +14 z^4 a^{-6} +11 z^4+11 a z^3+14 z^3 a^{-1} +5 z^3 a^{-3} +5 z^3 a^{-5} +3 z^3 a^{-7} +4 a^2 z^2-19 z^2 a^{-2} -20 z^2 a^{-4} -6 z^2 a^{-6} -z^2-2 a z-3 z a^{-1} -2 z a^{-3} -2 z a^{-5} -z a^{-7} -a^2+2 a^{-2} +3 a^{-4} + a^{-6}
The A2 invariant Data:K11a303/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a303/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (-1, 1)
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
-4 8 8 \frac{178}{3} \frac{86}{3} -32 \frac{80}{3} -\frac{64}{3} 40 -\frac{32}{3} 32 -\frac{712}{3} -\frac{344}{3} -\frac{4831}{30} \frac{22}{15} -\frac{12902}{45} \frac{2815}{18} -\frac{1951}{30}

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=0 is the signature of K11a303. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
15           1-1
13          3 3
11         61 -5
9        93  6
7       116   -5
5      139    4
3     1111     0
1    1013      -3
-1   712       5
-3  39        -6
-5 17         6
-7 3          -3
-91           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\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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K11a302.gif

K11a302

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K11a304