K11a38

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K11a37.gif

K11a37

K11a39.gif

K11a39

Contents

K11a38.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a38 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,5,15,6 X2837 X18,9,19,10 X16,11,17,12 X20,14,21,13 X6,15,7,16 X10,17,11,18 X22,19,1,20 X12,22,13,21
Gauss code 1, -4, 2, -1, 3, -8, 4, -2, 5, -9, 6, -11, 7, -3, 8, -6, 9, -5, 10, -7, 11, -10
Dowker-Thistlethwaite code 4 8 14 2 18 16 20 6 10 22 12
A Braid Representative
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A Morse Link Presentation K11a38 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a38/ThurstonBennequinNumber
Hyperbolic Volume 15.6826
A-Polynomial See Data:K11a38/A-polynomial

[edit Notes for K11a38's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,3]
Rasmussen s-Invariant 0

[edit Notes for K11a38's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-27 t+37-27 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 117, 0 }
Jones polynomial -q^5+3 q^4-7 q^3+13 q^2-16 q+19-19 q^{-1} +16 q^{-2} -12 q^{-3} +7 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) -a^2 z^6-z^6+a^4 z^4-3 a^2 z^4+2 z^4 a^{-2} -z^4+2 a^4 z^2-6 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +z^2+2 a^4-4 a^2+2 a^{-2} - a^{-4} +2
Kauffman polynomial (db, data sources) a^2 z^{10}+z^{10}+4 a^3 z^9+8 a z^9+4 z^9 a^{-1} +5 a^4 z^8+12 a^2 z^8+6 z^8 a^{-2} +13 z^8+3 a^5 z^7-4 a^3 z^7-10 a z^7+2 z^7 a^{-1} +5 z^7 a^{-3} +a^6 z^6-13 a^4 z^6-38 a^2 z^6-7 z^6 a^{-2} +3 z^6 a^{-4} -34 z^6-8 a^5 z^5-6 a^3 z^5-2 a z^5-11 z^5 a^{-1} -6 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+12 a^4 z^4+43 a^2 z^4+3 z^4 a^{-2} -5 z^4 a^{-4} +36 z^4+6 a^5 z^3+7 a^3 z^3+5 a z^3+7 z^3 a^{-1} +z^3 a^{-3} -2 z^3 a^{-5} +2 a^6 z^2-7 a^4 z^2-23 a^2 z^2+3 z^2 a^{-4} -17 z^2-2 a^5 z-2 a^3 z+z a^{-3} +z a^{-5} +2 a^4+4 a^2-2 a^{-2} - a^{-4} +2
The A2 invariant Data:K11a38/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a38/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a8, K11a187, K11a249,}

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

Vassiliev invariants

V2 and V3: (-1, 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
-4 24 8 \frac{34}{3} \frac{38}{3} -96 -208 -64 -72 -\frac{32}{3} 288 -\frac{136}{3} -\frac{152}{3} \frac{17489}{30} \frac{662}{15} \frac{6778}{45} \frac{1615}{18} -\frac{751}{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 K11a38. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-6-5-4-3-2-1012345χ
11           1-1
9          2 2
7         51 -4
5        82  6
3       85   -3
1      118    3
-1     99     0
-3    710      -3
-5   59       4
-7  27        -5
-9 15         4
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

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

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K11a37.gif

K11a37

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K11a39