K11a39

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K11a38.gif

K11a38

K11a40.gif

K11a40

Contents

K11a39.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a39 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X14,6,15,5 X2837 X18,9,19,10 X20,11,21,12 X6,14,7,13 X22,16,1,15 X12,17,13,18 X10,19,11,20 X16,22,17,21
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -9, 7, -3, 8, -11, 9, -5, 10, -6, 11, -8
Dowker-Thistlethwaite code 4 8 14 2 18 20 6 22 12 10 16
A Braid Representative
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A Morse Link Presentation K11a39 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a39/ThurstonBennequinNumber
Hyperbolic Volume 14.0334
A-Polynomial See Data:K11a39/A-polynomial

[edit Notes for K11a39's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a39's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+12 t^2-22 t+27-22 t^{-1} +12 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-6 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 101, 0 }
Jones polynomial -q^7+3 q^6-6 q^5+10 q^4-13 q^3+16 q^2-16 q+14-11 q^{-1} +7 q^{-2} -3 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) -2 z^6 a^{-2} -z^6+a^2 z^4-8 z^4 a^{-2} +3 z^4 a^{-4} -2 z^4+2 a^2 z^2-11 z^2 a^{-2} +9 z^2 a^{-4} -z^2 a^{-6} +a^2-5 a^{-2} +6 a^{-4} -2 a^{-6} +1
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +4 z^9 a^{-1} +7 z^9 a^{-3} +3 z^9 a^{-5} +11 z^8 a^{-2} +7 z^8 a^{-4} +3 z^8 a^{-6} +7 z^8+8 a z^7+2 z^7 a^{-1} -14 z^7 a^{-3} -7 z^7 a^{-5} +z^7 a^{-7} +6 a^2 z^6-36 z^6 a^{-2} -34 z^6 a^{-4} -12 z^6 a^{-6} -8 z^6+3 a^3 z^5-12 a z^5-18 z^5 a^{-1} -2 z^5 a^{-3} -3 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4-7 a^2 z^4+37 z^4 a^{-2} +43 z^4 a^{-4} +15 z^4 a^{-6} +z^4-2 a^3 z^3+10 a z^3+15 z^3 a^{-1} +10 z^3 a^{-3} +12 z^3 a^{-5} +5 z^3 a^{-7} -a^4 z^2+5 a^2 z^2-22 z^2 a^{-2} -24 z^2 a^{-4} -7 z^2 a^{-6} +z^2-3 a z-5 z a^{-1} -4 z a^{-3} -4 z a^{-5} -2 z a^{-7} -a^2+5 a^{-2} +6 a^{-4} +2 a^{-6} +1
The A2 invariant Data:K11a39/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a39/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, 2)
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 16 8 \frac{322}{3} \frac{158}{3} -64 \frac{160}{3} -\frac{32}{3} 48 -\frac{32}{3} 128 -\frac{1288}{3} -\frac{632}{3} -\frac{4591}{30} \frac{4222}{15} -\frac{26942}{45} \frac{2671}{18} -\frac{5551}{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 K11a39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
15           1-1
13          2 2
11         41 -3
9        62  4
7       74   -3
5      96    3
3     77     0
1    79      -2
-1   58       3
-3  26        -4
-5 15         4
-7 2          -2
-91           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{7}
r=1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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K11a38.gif

K11a38

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K11a40