K11a144

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K11a143.gif

K11a143

K11a145.gif

K11a145

Contents

K11a144.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a144 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 X6,17,7,18 X8,19,9,20 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -9, 4, -10, 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 6 8 14
A Braid Representative
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A Morse Link Presentation K11a144 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 3
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a144/ThurstonBennequinNumber
Hyperbolic Volume 11.8229
A-Polynomial See Data:K11a144/A-polynomial

[edit Notes for K11a144's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 4

[edit Notes for K11a144's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+10 t^2-16 t+17-16 t^{-1} +10 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-2 z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 73, -4 }
Jones polynomial 1-2 q^{-1} +4 q^{-2} -7 q^{-3} +10 q^{-4} -11 q^{-5} +12 q^{-6} -10 q^{-7} +8 q^{-8} -5 q^{-9} +2 q^{-10} - q^{-11}
HOMFLY-PT polynomial (db, data sources) -z^2 a^{10}-2 a^{10}+2 z^4 a^8+5 z^2 a^8+2 a^8-z^6 a^6-2 z^4 a^6+z^2 a^6+a^6-z^6 a^4-3 z^4 a^4-2 z^2 a^4-a^4+z^4 a^2+3 z^2 a^2+a^2
Kauffman polynomial (db, data sources) z^5 a^{13}-3 z^3 a^{13}+2 z a^{13}+2 z^6 a^{12}-4 z^4 a^{12}+z^2 a^{12}+3 z^7 a^{11}-6 z^5 a^{11}+4 z^3 a^{11}-2 z a^{11}+3 z^8 a^{10}-6 z^6 a^{10}+8 z^4 a^{10}-7 z^2 a^{10}+2 a^{10}+2 z^9 a^9-2 z^7 a^9+5 z^3 a^9-3 z a^9+z^{10} a^8-z^6 a^8+5 z^4 a^8-4 z^2 a^8+2 a^8+4 z^9 a^7-12 z^7 a^7+19 z^5 a^7-11 z^3 a^7+z a^7+z^{10} a^6-z^8 a^6+2 z^6 a^6-5 z^4 a^6+4 z^2 a^6-a^6+2 z^9 a^5-5 z^7 a^5+5 z^5 a^5-3 z^3 a^5-z a^5+2 z^8 a^4-4 z^6 a^4-2 z^4 a^4+4 z^2 a^4-a^4+2 z^7 a^3-7 z^5 a^3+6 z^3 a^3-z a^3+z^6 a^2-4 z^4 a^2+4 z^2 a^2-a^2
The A2 invariant Data:K11a144/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a144/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: (6, -17)
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
24 -136 288 892 156 -3264 -\frac{19600}{3} -\frac{3328}{3} -1096 2304 9248 21408 3744 \frac{240671}{5} -\frac{16972}{15} \frac{331364}{15} \frac{1729}{3} \frac{15471}{5}

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=-4 is the signature of K11a144. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-9-8-7-6-5-4-3-2-1012χ
1           11
-1          1 -1
-3         31 2
-5        52  -3
-7       52   3
-9      65    -1
-11     65     1
-13    46      2
-15   46       -2
-17  14        3
-19 14         -3
-21 1          1
-231           -1
Integral Khovanov Homology

(db, data source)

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