K11a43

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K11a42.gif

K11a42

K11a44.gif

K11a44

Contents

K11a43.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a43 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,6,15,5 X2837 X20,10,21,9 X16,12,17,11 X6,14,7,13 X18,16,19,15 X12,18,13,17 X22,20,1,19 X10,22,11,21
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -9, 7, -3, 8, -6, 9, -8, 10, -5, 11, -10
Dowker-Thistlethwaite code 4 8 14 2 20 16 6 18 12 22 10
A Braid Representative
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A Morse Link Presentation K11a43 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 3
Bridge index 4
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a43/ThurstonBennequinNumber
Hyperbolic Volume 16.3093
A-Polynomial See Data:K11a43/A-polynomial

[edit Notes for K11a43's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11a43's four dimensional invariants]

Polynomial invariants

Alexander polynomial 4 t^3-15 t^2+30 t-37+30 t^{-1} -15 t^{-2} +4 t^{-3}
Conway polynomial 4 z^6+9 z^4+6 z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 135, 6 }
Jones polynomial -q^{14}+4 q^{13}-8 q^{12}+14 q^{11}-20 q^{10}+21 q^9-22 q^8+19 q^7-13 q^6+9 q^5-3 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +3 z^6 a^{-8} +3 z^4 a^{-6} +12 z^4 a^{-8} -6 z^4 a^{-10} +3 z^2 a^{-6} +16 z^2 a^{-8} -17 z^2 a^{-10} +4 z^2 a^{-12} + a^{-6} +7 a^{-8} -12 a^{-10} +6 a^{-12} - a^{-14}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +4 z^9 a^{-9} +8 z^9 a^{-11} +4 z^9 a^{-13} +6 z^8 a^{-8} +16 z^8 a^{-10} +17 z^8 a^{-12} +7 z^8 a^{-14} +3 z^7 a^{-7} +4 z^7 a^{-9} +7 z^7 a^{-11} +13 z^7 a^{-13} +7 z^7 a^{-15} +z^6 a^{-6} -15 z^6 a^{-8} -41 z^6 a^{-10} -31 z^6 a^{-12} -2 z^6 a^{-14} +4 z^6 a^{-16} -6 z^5 a^{-7} -30 z^5 a^{-9} -54 z^5 a^{-11} -41 z^5 a^{-13} -10 z^5 a^{-15} +z^5 a^{-17} -3 z^4 a^{-6} +18 z^4 a^{-8} +38 z^4 a^{-10} +12 z^4 a^{-12} -11 z^4 a^{-14} -6 z^4 a^{-16} +3 z^3 a^{-7} +35 z^3 a^{-9} +63 z^3 a^{-11} +39 z^3 a^{-13} +7 z^3 a^{-15} -z^3 a^{-17} +3 z^2 a^{-6} -16 z^2 a^{-8} -27 z^2 a^{-10} -3 z^2 a^{-12} +8 z^2 a^{-14} +3 z^2 a^{-16} -15 z a^{-9} -27 z a^{-11} -15 z a^{-13} -3 z a^{-15} - a^{-6} +7 a^{-8} +12 a^{-10} +6 a^{-12} + a^{-14}
The A2 invariant Data:K11a43/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a43/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, 12)
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 96 288 524 68 2304 3232 544 320 2304 4608 12576 1632 \frac{103671}{5} \frac{23708}{15} \frac{93004}{15} \frac{233}{3} \frac{3671}{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=6 is the signature of K11a43. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          3 3
25         51 -4
23        93  6
21       115   -6
19      109    1
17     1211     -1
15    710      -3
13   612       6
11  37        -4
9  6         6
713          -2
51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{3}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=6 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=7 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=8 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=9 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=11 {\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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K11a42

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K11a44