K11a41

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K11a40.gif

K11a40

K11a42.gif

K11a42

Contents

K11a41.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a41 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X18,10,19,9 X20,12,21,11 X16,13,17,14 X6,15,7,16 X22,18,1,17 X12,20,13,19 X10,22,11,21
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -10, 7, -3, 8, -7, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code 4 8 14 2 18 20 16 6 22 12 10
A Braid Representative
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A Morse Link Presentation K11a41 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a41/ThurstonBennequinNumber
Hyperbolic Volume 14.6854
A-Polynomial See Data:K11a41/A-polynomial

[edit Notes for K11a41'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 K11a41's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-11 t^2+27 t-35+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) \{1\}
Determinant and Signature { 115, 2 }
Jones polynomial q^9-3 q^8+6 q^7-11 q^6+15 q^5-18 q^4+19 q^3-16 q^2+13 q-8+4 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +2 z^4 a^{-2} +2 z^4 a^{-4} -2 z^4 a^{-6} -z^4+2 z^2 a^{-2} +3 z^2 a^{-4} -4 z^2 a^{-6} +z^2 a^{-8} -z^2+ a^{-2} +2 a^{-4} -3 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) z^{10} a^{-4} +z^{10} a^{-6} +4 z^9 a^{-3} +7 z^9 a^{-5} +3 z^9 a^{-7} +7 z^8 a^{-2} +12 z^8 a^{-4} +9 z^8 a^{-6} +4 z^8 a^{-8} +7 z^7 a^{-1} +4 z^7 a^{-3} -6 z^7 a^{-5} +3 z^7 a^{-9} -8 z^6 a^{-2} -29 z^6 a^{-4} -27 z^6 a^{-6} -9 z^6 a^{-8} +z^6 a^{-10} +4 z^6+a z^5-11 z^5 a^{-1} -18 z^5 a^{-3} -12 z^5 a^{-5} -15 z^5 a^{-7} -9 z^5 a^{-9} +26 z^4 a^{-4} +28 z^4 a^{-6} +5 z^4 a^{-8} -3 z^4 a^{-10} -6 z^4-a z^3+4 z^3 a^{-1} +14 z^3 a^{-3} +20 z^3 a^{-5} +19 z^3 a^{-7} +8 z^3 a^{-9} +z^2 a^{-2} -10 z^2 a^{-4} -13 z^2 a^{-6} -2 z^2 a^{-8} +2 z^2 a^{-10} +2 z^2-z a^{-1} -3 z a^{-3} -7 z a^{-5} -8 z a^{-7} -3 z a^{-9} - a^{-2} +2 a^{-4} +3 a^{-6} + a^{-8}
The A2 invariant Data:K11a41/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a41/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_121, K11a183, K11a198, K11a331,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a183,}

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{82}{3} -\frac{14}{3} 0 -96 0 0 \frac{32}{3} 0 -\frac{328}{3} -\frac{56}{3} -\frac{7649}{30} -\frac{714}{5} \frac{8222}{45} -\frac{223}{18} \frac{991}{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 K11a41. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
19           11
17          2 -2
15         41 3
13        72  -5
11       84   4
9      107    -3
7     98     1
5    710      3
3   69       -3
1  38        5
-1 15         -4
-3 3          3
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=8 {\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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K11a40.gif

K11a40

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K11a42