K11n124

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K11n123.gif

K11n123

K11n125.gif

K11n125

Contents

K11n124.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n124 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,6,17,5 X7,15,8,14 X9,19,10,18 X2,11,3,12 X13,9,14,8 X20,15,21,16 X22,18,1,17 X19,13,20,12 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, -4, 7, -5, -2, 6, 10, -7, 4, 8, -3, 9, 5, -10, -8, 11, -9
Dowker-Thistlethwaite code 4 10 16 -14 -18 2 -8 20 22 -12 6
A Braid Representative
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A Morse Link Presentation K11n124 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n124/ThurstonBennequinNumber
Hyperbolic Volume 13.6991
A-Polynomial See Data:K11n124/A-polynomial

[edit Notes for K11n124'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 K11n124's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-6 t^2+14 t-17+14 t^{-1} -6 t^{-2} + t^{-3}
Conway polynomial z^6-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 59, 2 }
Jones polynomial -q^6+4 q^5-7 q^4+9 q^3-10 q^2+10 q-8+6 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +3 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} -4 z^2+a^2+ a^{-2} -1
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +5 z^8 a^{-2} +2 z^8 a^{-4} +3 z^8+3 a z^7+5 z^7 a^{-1} +3 z^7 a^{-3} +z^7 a^{-5} +a^2 z^6-7 z^6 a^{-2} -z^6 a^{-4} -5 z^6-9 a z^5-18 z^5 a^{-1} -6 z^5 a^{-3} +3 z^5 a^{-5} -3 a^2 z^4-4 z^4 a^{-2} +2 z^4 a^{-4} +4 z^4 a^{-6} -5 z^4+7 a z^3+10 z^3 a^{-1} +z^3 a^{-3} -z^3 a^{-5} +z^3 a^{-7} +3 a^2 z^2+4 z^2 a^{-2} -z^2 a^{-4} -z^2 a^{-6} +7 z^2-a z-z a^{-1} -a^2- a^{-2} -1
The A2 invariant Data:K11n124/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n124/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_32, K11n52,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n166,}

Vassiliev invariants

V2 and V3: (-1, 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
-4 0 8 \frac{34}{3} \frac{14}{3} 0 0 -32 32 -\frac{32}{3} 0 -\frac{136}{3} -\frac{56}{3} -\frac{1231}{30} -\frac{1778}{15} \frac{4018}{45} \frac{79}{18} \frac{689}{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=2 is the signature of K11n124. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-1012345χ
13         1-1
11        3 3
9       41 -3
7      53  2
5     54   -1
3    55    0
1   46     2
-1  24      -2
-3 14       3
-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}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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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K11n123.gif

K11n123

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K11n125