K11n143

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K11n142.gif

K11n142

K11n144.gif

K11n144

Contents

K11n143.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n143 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X5,16,6,17 X22,8,1,7 X9,18,10,19 X2,12,3,11 X13,21,14,20 X15,10,16,11 X17,6,18,7 X19,15,20,14 X8,22,9,21
Gauss code 1, -6, 2, -1, -3, 9, 4, -11, -5, 8, 6, -2, -7, 10, -8, 3, -9, 5, -10, 7, 11, -4
Dowker-Thistlethwaite code 4 12 -16 22 -18 2 -20 -10 -6 -14 8
A Braid Representative
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A Morse Link Presentation K11n143 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:K11n143/ThurstonBennequinNumber
Hyperbolic Volume 9.31377
A-Polynomial See Data:K11n143/A-polynomial

[edit Notes for K11n143's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11n143's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+2 t^2-t+1- t^{-1} +2 t^{-2} - t^{-3}
Conway polynomial -z^6-4 z^4-2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 9, 0 }
Jones polynomial q^6-2 q^5+2 q^4-2 q^3+2 q^2-q+1- q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -5 z^4 a^{-2} +z^4 a^{-4} +a^2 z^2-6 z^2 a^{-2} +3 z^2 a^{-4} +a^2- a^{-2} + a^{-4}
Kauffman polynomial (db, data sources) z^8 a^{-2} +z^8 a^{-4} +a z^7+z^7 a^{-1} +2 z^7 a^{-3} +2 z^7 a^{-5} +a^2 z^6-6 z^6 a^{-2} -4 z^6 a^{-4} +z^6 a^{-6} -6 a z^5-7 z^5 a^{-1} -10 z^5 a^{-3} -9 z^5 a^{-5} -5 a^2 z^4+10 z^4 a^{-2} +3 z^4 a^{-4} -4 z^4 a^{-6} -2 z^4+8 a z^3+11 z^3 a^{-1} +12 z^3 a^{-3} +9 z^3 a^{-5} +5 a^2 z^2-7 z^2 a^{-2} -2 z^2 a^{-4} +3 z^2 a^{-6} +3 z^2-2 a z-4 z a^{-1} -4 z a^{-3} -2 z a^{-5} -a^2+ a^{-2} + a^{-4}
The A2 invariant Data:K11n143/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n143/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: (-2, -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
-8 -8 32 \frac{212}{3} \frac{148}{3} 64 \frac{496}{3} \frac{160}{3} 56 -\frac{256}{3} 32 -\frac{1696}{3} -\frac{1184}{3} -\frac{5431}{15} \frac{5524}{15} -\frac{37924}{45} \frac{1543}{9} -\frac{3511}{15}

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 K11n143. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-10123456χ
13          11
11         1 -1
9        11 0
7      121  0
5      11   0
3    122    1
1   121     0
-1   12      1
-3 111       -1
-5           0
-71          1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}
r=-1 {\mathbb Z} {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=1 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=6 {\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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K11n142.gif

K11n142

K11n144.gif

K11n144