K11n144

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K11n143.gif

K11n143

K11n145.gif

K11n145

Contents

K11n144.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n144 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X18,6,19,5 X7,15,8,14 X16,10,17,9 X2,12,3,11 X22,13,1,14 X15,20,16,21 X10,18,11,17 X6,20,7,19 X21,9,22,8
Gauss code 1, -6, 2, -1, 3, -10, -4, 11, 5, -9, 6, -2, 7, 4, -8, -5, 9, -3, 10, 8, -11, -7
Dowker-Thistlethwaite code 4 12 18 -14 16 2 22 -20 10 6 -8
A Braid Representative
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A Morse Link Presentation K11n144 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n144/ThurstonBennequinNumber
Hyperbolic Volume 13.8021
A-Polynomial See Data:K11n144/A-polynomial

[edit Notes for K11n144'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 K11n144's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+7 t^2-15 t+19-15 t^{-1} +7 t^{-2} - t^{-3}
Conway polynomial -z^6+z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 65, 4 }
Jones polynomial -q^{11}+3 q^{10}-6 q^9+9 q^8-11 q^7+11 q^6-10 q^5+8 q^4-4 q^3+2 q^2
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-6} +2 z^4 a^{-4} -3 z^4 a^{-6} +2 z^4 a^{-8} +5 z^2 a^{-4} -4 z^2 a^{-6} +4 z^2 a^{-8} -z^2 a^{-10} +3 a^{-4} -3 a^{-6} +2 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^9 a^{-7} +z^9 a^{-9} +2 z^8 a^{-6} +6 z^8 a^{-8} +4 z^8 a^{-10} +z^7 a^{-5} +3 z^7 a^{-7} +7 z^7 a^{-9} +5 z^7 a^{-11} -3 z^6 a^{-6} -13 z^6 a^{-8} -7 z^6 a^{-10} +3 z^6 a^{-12} +z^5 a^{-5} -7 z^5 a^{-7} -21 z^5 a^{-9} -12 z^5 a^{-11} +z^5 a^{-13} +3 z^4 a^{-4} +8 z^4 a^{-6} +16 z^4 a^{-8} +5 z^4 a^{-10} -6 z^4 a^{-12} -z^3 a^{-5} +8 z^3 a^{-7} +21 z^3 a^{-9} +10 z^3 a^{-11} -2 z^3 a^{-13} -6 z^2 a^{-4} -10 z^2 a^{-6} -9 z^2 a^{-8} -3 z^2 a^{-10} +2 z^2 a^{-12} -2 z a^{-5} -4 z a^{-7} -6 z a^{-9} -4 z a^{-11} +3 a^{-4} +3 a^{-6} +2 a^{-8} + a^{-10}
The A2 invariant 2 q^{-6} - q^{-8} +3 q^{-10} + q^{-12} - q^{-14} +2 q^{-16} -3 q^{-18} + q^{-20} - q^{-22} +2 q^{-26} -2 q^{-28} + q^{-30} - q^{-34}
The G2 invariant Data:K11n144/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n10, K11n103,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n10,}

Vassiliev invariants

V2 and V3: (4, 9)
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
16 72 128 \frac{1160}{3} \frac{160}{3} 1152 2288 352 328 \frac{2048}{3} 2592 \frac{18560}{3} \frac{2560}{3} \frac{207542}{15} -\frac{928}{15} \frac{251408}{45} \frac{2074}{9} \frac{10742}{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=4 is the signature of K11n144. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
23         1-1
21        2 2
19       41 -3
17      52  3
15     64   -2
13    55    0
11   56     1
9  35      -2
7 15       4
513        -2
32         2
Integral Khovanov Homology

(db, data source)

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

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K11n143.gif

K11n143

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K11n145