K11a249

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K11a248.gif

K11a248

K11a250.gif

K11a250

Contents

K11a249.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a249 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X8394 X10,6,11,5 X18,8,19,7 X16,9,17,10 X22,11,1,12 X20,13,21,14 X4,16,5,15 X2,17,3,18 X14,19,15,20 X12,21,13,22
Gauss code 1, -9, 2, -8, 3, -1, 4, -2, 5, -3, 6, -11, 7, -10, 8, -5, 9, -4, 10, -7, 11, -6
Dowker-Thistlethwaite code 6 8 10 18 16 22 20 4 2 14 12
A Braid Representative
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A Morse Link Presentation K11a249 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a249/ThurstonBennequinNumber
Hyperbolic Volume 15.797
A-Polynomial See Data:K11a249/A-polynomial

[edit Notes for K11a249's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,3]
Rasmussen s-Invariant 0

[edit Notes for K11a249's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-27 t+37-27 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4-z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 117, 0 }
Jones polynomial q^4-4 q^3+8 q^2-12 q+17-18 q^{-1} +18 q^{-2} -16 q^{-3} +11 q^{-4} -7 q^{-5} +4 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6+2 z^4 a^4+3 z^2 a^4+a^4-z^6 a^2-2 z^4 a^2-3 z^2 a^2-2 a^2-z^6-2 z^4-z^2+2+z^4 a^{-2} +z^2 a^{-2}
Kauffman polynomial (db, data sources) 3 a^4 z^{10}+3 a^2 z^{10}+6 a^5 z^9+14 a^3 z^9+8 a z^9+4 a^6 z^8-a^4 z^8+5 a^2 z^8+10 z^8+a^7 z^7-23 a^5 z^7-49 a^3 z^7-15 a z^7+10 z^7 a^{-1} -15 a^6 z^6-26 a^4 z^6-33 a^2 z^6+8 z^6 a^{-2} -14 z^6-3 a^7 z^5+25 a^5 z^5+53 a^3 z^5+10 a z^5-11 z^5 a^{-1} +4 z^5 a^{-3} +14 a^6 z^4+37 a^4 z^4+35 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +3 z^4+a^7 z^3-8 a^5 z^3-18 a^3 z^3-6 a z^3+z^3 a^{-1} -2 z^3 a^{-3} -3 a^6 z^2-12 a^4 z^2-12 a^2 z^2+2 z^2 a^{-2} -z^2-a^5 z-a^3 z+a^4+2 a^2+2
The A2 invariant Data:K11a249/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a249/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a8, K11a38, K11a187,}

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{34}{3} \frac{38}{3} -32 -\frac{208}{3} \frac{128}{3} -88 -\frac{32}{3} 32 -\frac{136}{3} -\frac{152}{3} \frac{2129}{30} -\frac{2698}{15} \frac{3898}{45} \frac{2191}{18} \frac{209}{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 K11a249. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-101234χ
9           11
7          3 -3
5         51 4
3        73  -4
1       105   5
-1      98    -1
-3     99     0
-5    79      2
-7   49       -5
-9  37        4
-11 14         -3
-13 3          3
-151           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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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K11a248

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K11a250