K11a132

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K11a131.gif

K11a131

K11a133.gif

K11a133

Contents

K11a132.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a132 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,6,17,5 X12,8,13,7 X18,9,19,10 X2,11,3,12 X20,14,21,13 X6,16,7,15 X22,18,1,17 X8,19,9,20 X14,22,15,21
Gauss code 1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -4, 7, -11, 8, -3, 9, -5, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 10 16 12 18 2 20 6 22 8 14
A Braid Representative
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A Morse Link Presentation K11a132 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:K11a132/ThurstonBennequinNumber
Hyperbolic Volume 16.7788
A-Polynomial See Data:K11a132/A-polynomial

[edit Notes for K11a132'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 K11a132's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-13 t^2+32 t-41+32 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 { 135, 2 }
Jones polynomial -q^8+4 q^7-9 q^6+15 q^5-19 q^4+22 q^3-22 q^2+18 q-13+8 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2-4 z^2 a^{-2} +4 z^2 a^{-4} -z^2 a^{-6} -2 z^2+a^2-4 a^{-2} +4 a^{-4} - a^{-6} +1
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10} a^{-4} +5 z^9 a^{-1} +12 z^9 a^{-3} +7 z^9 a^{-5} +9 z^8 a^{-2} +14 z^8 a^{-4} +10 z^8 a^{-6} +5 z^8+3 a z^7-7 z^7 a^{-1} -23 z^7 a^{-3} -5 z^7 a^{-5} +8 z^7 a^{-7} +a^2 z^6-30 z^6 a^{-2} -40 z^6 a^{-4} -17 z^6 a^{-6} +4 z^6 a^{-8} -10 z^6-7 a z^5+2 z^5 a^{-1} +17 z^5 a^{-3} -5 z^5 a^{-5} -12 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+33 z^4 a^{-2} +42 z^4 a^{-4} +12 z^4 a^{-6} -5 z^4 a^{-8} +5 z^4+4 a z^3-4 z^3 a^{-1} -8 z^3 a^{-3} +6 z^3 a^{-5} +5 z^3 a^{-7} -z^3 a^{-9} +3 a^2 z^2-20 z^2 a^{-2} -21 z^2 a^{-4} -5 z^2 a^{-6} +z^2 a^{-8} -2 z^2+3 z a^{-1} +3 z a^{-3} -z a^{-5} -z a^{-7} -a^2+4 a^{-2} +4 a^{-4} + a^{-6} +1
The A2 invariant Data:K11a132/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a132/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a6, K11a352,}

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

Vassiliev invariants

V2 and V3: (-2, 0)
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 0 32 \frac{212}{3} \frac{76}{3} 0 32 -32 32 -\frac{256}{3} 0 -\frac{1696}{3} -\frac{608}{3} -\frac{9511}{15} -\frac{1796}{15} -\frac{14524}{45} \frac{391}{9} -\frac{871}{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 K11a132. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          3 3
13         61 -5
11        93  6
9       106   -4
7      129    3
5     1010     0
3    812      -4
1   611       5
-1  27        -5
-3 16         5
-5 2          -2
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
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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K11a131.gif

K11a131

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K11a133