K11a145

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K11a144.gif

K11a144

K11a146.gif

K11a146

Contents

K11a145.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a145 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,5,17,6 X18,7,19,8 X12,10,13,9 X2,11,3,12 X22,13,1,14 X20,15,21,16 X8,17,9,18 X6,19,7,20 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -5, 7, -11, 8, -3, 9, -4, 10, -8, 11, -7
Dowker-Thistlethwaite code 4 10 16 18 12 2 22 20 8 6 14
A Braid Representative
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A Morse Link Presentation K11a145 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a145/ThurstonBennequinNumber
Hyperbolic Volume 12.3911
A-Polynomial See Data:K11a145/A-polynomial

[edit Notes for K11a145'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 K11a145's four dimensional invariants]

Polynomial invariants

Alexander polynomial -4 t^2+21 t-33+21 t^{-1} -4 t^{-2}
Conway polynomial -4 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 83, -2 }
Jones polynomial q-3+6 q^{-1} -9 q^{-2} +12 q^{-3} -13 q^{-4} +13 q^{-5} -10 q^{-6} +8 q^{-7} -5 q^{-8} +2 q^{-9} - q^{-10}
HOMFLY-PT polynomial (db, data sources) -a^{10}+2 z^2 a^8-z^4 a^6+2 z^2 a^6+2 a^6-2 z^4 a^4-z^2 a^4-a^4-z^4 a^2+z^2 a^2+a^2+z^2
Kauffman polynomial (db, data sources) z^7 a^{11}-5 z^5 a^{11}+8 z^3 a^{11}-4 z a^{11}+2 z^8 a^{10}-8 z^6 a^{10}+9 z^4 a^{10}-3 z^2 a^{10}+a^{10}+2 z^9 a^9-5 z^7 a^9-z^5 a^9+6 z^3 a^9-2 z a^9+z^{10} a^8+z^8 a^8-9 z^6 a^8+5 z^4 a^8+z^2 a^8+5 z^9 a^7-13 z^7 a^7+10 z^5 a^7-7 z^3 a^7+2 z a^7+z^{10} a^6+4 z^8 a^6-12 z^6 a^6+4 z^4 a^6+3 z^2 a^6-2 a^6+3 z^9 a^5-z^7 a^5-6 z^5 a^5+5 z^3 a^5-z a^5+5 z^8 a^4-6 z^6 a^4+z^4 a^4+3 z^2 a^4-a^4+6 z^7 a^3-9 z^5 a^3+7 z^3 a^3-z a^3+5 z^6 a^2-6 z^4 a^2+3 z^2 a^2-a^2+3 z^5 a-3 z^3 a+z^4-z^2
The A2 invariant Data:K11a145/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a145/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (5, -13)
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
20 -104 200 \frac{1942}{3} \frac{386}{3} -2080 -\frac{13136}{3} -\frac{2144}{3} -872 \frac{4000}{3} 5408 \frac{38840}{3} \frac{7720}{3} \frac{177391}{6} -2098 \frac{139526}{9} \frac{8053}{18} \frac{13999}{6}

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 K11a145. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-1012χ
3           11
1          2 -2
-1         41 3
-3        63  -3
-5       63   3
-7      76    -1
-9     66     0
-11    47      3
-13   46       -2
-15  14        3
-17 14         -3
-19 1          1
-211           -1
Integral Khovanov Homology

(db, data source)

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