K11a351

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K11a350.gif

K11a350

K11a352.gif

K11a352

Contents

K11a351.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a351 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X18,4,19,3 X16,5,17,6 X14,8,15,7 X20,9,21,10 X4,12,5,11 X2,13,3,14 X22,16,1,15 X12,18,13,17 X8,19,9,20 X10,21,11,22
Gauss code 1, -7, 2, -6, 3, -1, 4, -10, 5, -11, 6, -9, 7, -4, 8, -3, 9, -2, 10, -5, 11, -8
Dowker-Thistlethwaite code 6 18 16 14 20 4 2 22 12 8 10
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart4.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a351 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a351/ThurstonBennequinNumber
Hyperbolic Volume 18.5253
A-Polynomial See Data:K11a351/A-polynomial

[edit Notes for K11a351's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a351's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-7 t^3+20 t^2-34 t+41-34 t^{-1} +20 t^{-2} -7 t^{-3} + t^{-4}
Conway polynomial z^8+z^6-2 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 165, 0 }
Jones polynomial q^6-4 q^5+9 q^4-16 q^3+23 q^2-26 q+27-24 q^{-1} +18 q^{-2} -11 q^{-3} +5 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +4 z^6-2 a^2 z^4-6 z^4 a^{-2} +z^4 a^{-4} +5 z^4-4 z^2 a^{-2} +2 z^2 a^{-4} +z^2+a^2+ a^{-2} -1
Kauffman polynomial (db, data sources) 4 z^{10} a^{-2} +4 z^{10}+11 a z^9+20 z^9 a^{-1} +9 z^9 a^{-3} +14 a^2 z^8+8 z^8 a^{-2} +8 z^8 a^{-4} +14 z^8+11 a^3 z^7-10 a z^7-45 z^7 a^{-1} -20 z^7 a^{-3} +4 z^7 a^{-5} +5 a^4 z^6-19 a^2 z^6-37 z^6 a^{-2} -20 z^6 a^{-4} +z^6 a^{-6} -40 z^6+a^5 z^5-14 a^3 z^5-3 a z^5+37 z^5 a^{-1} +16 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+4 a^2 z^4+38 z^4 a^{-2} +17 z^4 a^{-4} -2 z^4 a^{-6} +27 z^4+3 a^3 z^3-a z^3-13 z^3 a^{-1} -5 z^3 a^{-3} +4 z^3 a^{-5} +a^2 z^2-11 z^2 a^{-2} -6 z^2 a^{-4} -4 z^2+a^3 z+2 a z+2 z a^{-1} +z a^{-3} -a^2- a^{-2} -1
The A2 invariant -q^{14}+3 q^{12}-3 q^{10}+3 q^8+q^6-4 q^4+5 q^2-6+4 q^{-2} - q^{-4} +5 q^{-8} -4 q^{-10} +2 q^{-12} - q^{-14} - q^{-16} + q^{-18}
The G2 invariant Data:K11a351/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: (-1, 0)
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
-4 0 8 \frac{82}{3} \frac{62}{3} 0 0 64 -64 -\frac{32}{3} 0 -\frac{328}{3} -\frac{248}{3} -\frac{2911}{30} \frac{1262}{15} -\frac{8462}{45} \frac{415}{18} -\frac{991}{30}

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 K11a351. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         61 5
7        103  -7
5       136   7
3      1310    -3
1     1413     1
-1    1114      3
-3   713       -6
-5  411        7
-7 17         -6
-9 4          4
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{14}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a350.gif

K11a350

K11a352.gif

K11a352