K11n43

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K11n42.gif

K11n42

K11n44.gif

K11n44

Contents

K11n43.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n43 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,13,6,12 X2837 X18,10,19,9 X20,11,21,12 X13,7,14,6 X10,16,11,15 X22,18,1,17 X14,19,15,20 X16,22,17,21
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, 5, -8, 6, 3, -7, -10, 8, -11, 9, -5, 10, -6, 11, -9
Dowker-Thistlethwaite code 4 8 -12 2 18 20 -6 10 22 14 16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11n43 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n43/ThurstonBennequinNumber
Hyperbolic Volume 15.7945
A-Polynomial See Data:K11n43/A-polynomial

[edit Notes for K11n43'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 K11n43's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+10 t^2-20 t+25-20 t^{-1} +10 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-2 z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 89, 4 }
Jones polynomial -q^{11}+4 q^{10}-8 q^9+12 q^8-15 q^7+15 q^6-14 q^5+11 q^4-6 q^3+3 q^2
HOMFLY-PT polynomial (db, data sources) -2 z^6 a^{-6} +3 z^4 a^{-4} -8 z^4 a^{-6} +3 z^4 a^{-8} +8 z^2 a^{-4} -12 z^2 a^{-6} +7 z^2 a^{-8} -z^2 a^{-10} +5 a^{-4} -7 a^{-6} +4 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) 2 z^9 a^{-7} +2 z^9 a^{-9} +5 z^8 a^{-6} +11 z^8 a^{-8} +6 z^8 a^{-10} +3 z^7 a^{-5} +7 z^7 a^{-7} +11 z^7 a^{-9} +7 z^7 a^{-11} -11 z^6 a^{-6} -20 z^6 a^{-8} -5 z^6 a^{-10} +4 z^6 a^{-12} -3 z^5 a^{-5} -20 z^5 a^{-7} -30 z^5 a^{-9} -12 z^5 a^{-11} +z^5 a^{-13} +6 z^4 a^{-4} +21 z^4 a^{-6} +16 z^4 a^{-8} -5 z^4 a^{-10} -6 z^4 a^{-12} +4 z^3 a^{-5} +20 z^3 a^{-7} +23 z^3 a^{-9} +6 z^3 a^{-11} -z^3 a^{-13} -11 z^2 a^{-4} -21 z^2 a^{-6} -10 z^2 a^{-8} +2 z^2 a^{-10} +2 z^2 a^{-12} -4 z a^{-5} -8 z a^{-7} -6 z a^{-9} -2 z a^{-11} +5 a^{-4} +7 a^{-6} +4 a^{-8} + a^{-10}
The A2 invariant 3 q^{-6} - q^{-8} +4 q^{-10} + q^{-12} -2 q^{-14} +2 q^{-16} -5 q^{-18} + q^{-20} -2 q^{-22} +3 q^{-26} -2 q^{-28} +2 q^{-30} - q^{-34}
The G2 invariant Data:K11n43/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_92, K11a153, K11a224, K11n35,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n35,}

Vassiliev invariants

V2 and V3: (2, 3)
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 24 32 \frac{364}{3} \frac{92}{3} 192 592 128 120 \frac{256}{3} 288 \frac{2912}{3} \frac{736}{3} \frac{39991}{15} -\frac{268}{5} \frac{62644}{45} \frac{377}{9} \frac{2551}{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 K11n43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
23         1-1
21        3 3
19       51 -4
17      73  4
15     85   -3
13    77    0
11   78     1
9  47      -3
7 27       5
514        -3
33         3
Integral Khovanov Homology

(db, data source)

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

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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K11n42.gif

K11n42

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K11n44