K11a42

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K11a41.gif

K11a41

K11a43.gif

K11a43

Contents

K11a42.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a42 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,6,15,5 X2837 X20,9,21,10 X16,11,17,12 X6,14,7,13 X12,15,13,16 X22,18,1,17 X10,19,11,20 X18,22,19,21
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -8, 7, -3, 8, -6, 9, -11, 10, -5, 11, -9
Dowker-Thistlethwaite code 4 8 14 2 20 16 6 12 22 10 18
A Braid Representative
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A Morse Link Presentation K11a42 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:K11a42/ThurstonBennequinNumber
Hyperbolic Volume 14.5336
A-Polynomial See Data:K11a42/A-polynomial

[edit Notes for K11a42'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 K11a42's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-9 t^2+26 t-35+26 t^{-1} -9 t^{-2} + t^{-3}
Conway polynomial z^6-3 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 107, 2 }
Jones polynomial -q^8+3 q^7-6 q^6+11 q^5-15 q^4+17 q^3-17 q^2+15 q-11+7 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +2 z^4 a^{-2} -3 z^4 a^{-4} -2 z^4+a^2 z^2+3 z^2 a^{-2} -5 z^2 a^{-4} +3 z^2 a^{-6} -3 z^2+a^2+2 a^{-2} -3 a^{-4} +3 a^{-6} - a^{-8} -1
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +6 z^9 a^{-3} +3 z^9 a^{-5} +7 z^8 a^{-2} +7 z^8 a^{-4} +4 z^8 a^{-6} +4 z^8+3 a z^7-5 z^7 a^{-3} +2 z^7 a^{-5} +4 z^7 a^{-7} +a^2 z^6-15 z^6 a^{-2} -10 z^6 a^{-4} +3 z^6 a^{-8} -7 z^6-8 a z^5-11 z^5 a^{-1} -5 z^5 a^{-3} -7 z^5 a^{-5} -4 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+4 z^4 a^{-2} -2 z^4 a^{-4} -10 z^4 a^{-6} -6 z^4 a^{-8} -z^4+6 a z^3+7 z^3 a^{-1} +5 z^3 a^{-3} +5 z^3 a^{-5} -z^3 a^{-7} -2 z^3 a^{-9} +3 a^2 z^2+3 z^2 a^{-2} +9 z^2 a^{-4} +11 z^2 a^{-6} +4 z^2 a^{-8} +4 z^2-a z-z a^{-1} -z a^{-3} -z a^{-5} +z a^{-7} +z a^{-9} -a^2-2 a^{-2} -3 a^{-4} -3 a^{-6} - a^{-8} -1
The A2 invariant Data:K11a42/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a42/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a10,}

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{226}{3} \frac{86}{3} -32 \frac{464}{3} \frac{32}{3} 72 -\frac{32}{3} 32 -\frac{904}{3} -\frac{344}{3} \frac{2129}{30} -\frac{286}{5} \frac{778}{45} \frac{1231}{18} -\frac{751}{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 K11a42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          2 2
13         41 -3
11        72  5
9       84   -4
7      97    2
5     88     0
3    79      -2
1   59       4
-1  26        -4
-3 15         4
-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}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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K11a41.gif

K11a41

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K11a43