K11a13

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K11a12.gif

K11a12

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K11a14

Contents

K11a13.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a13 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X16,7,17,8 X2,9,3,10 X22,12,1,11 X20,14,21,13 X18,16,19,15 X6,17,7,18 X14,20,15,19 X12,22,13,21
Gauss code 1, -5, 2, -1, 3, -9, 4, -2, 5, -3, 6, -11, 7, -10, 8, -4, 9, -8, 10, -7, 11, -6
Dowker-Thistlethwaite code 4 8 10 16 2 22 20 18 6 14 12
A Braid Representative
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A Morse Link Presentation K11a13 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a13/ThurstonBennequinNumber
Hyperbolic Volume 10.7354
A-Polynomial See Data:K11a13/A-polynomial

[edit Notes for K11a13's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant 0

[edit Notes for K11a13's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^2-15 t+25-15 t^{-1} +3 t^{-2}
Conway polynomial 3 z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 61, 0 }
Jones polynomial -q^7+3 q^6-4 q^5+6 q^4-8 q^3+9 q^2-9 q+8-6 q^{-1} +4 q^{-2} -2 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) a^4-2 z^2 a^2-a^2+z^4+1+z^4 a^{-2} -z^2 a^{-2} - a^{-2} +z^4 a^{-4} +z^2 a^{-4} + a^{-4} -z^2 a^{-6}
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +5 z^9 a^{-3} +3 z^9 a^{-5} +z^8 a^{-4} +3 z^8 a^{-6} +2 z^8+2 a z^7-5 z^7 a^{-1} -21 z^7 a^{-3} -13 z^7 a^{-5} +z^7 a^{-7} +2 a^2 z^6-8 z^6 a^{-2} -18 z^6 a^{-4} -14 z^6 a^{-6} -2 z^6+2 a^3 z^5+8 z^5 a^{-1} +29 z^5 a^{-3} +15 z^5 a^{-5} -4 z^5 a^{-7} +a^4 z^4+13 z^4 a^{-2} +27 z^4 a^{-4} +17 z^4 a^{-6} +2 z^4-3 a^3 z^3-2 a z^3-8 z^3 a^{-1} -18 z^3 a^{-3} -6 z^3 a^{-5} +3 z^3 a^{-7} -2 a^4 z^2-3 a^2 z^2-8 z^2 a^{-2} -12 z^2 a^{-4} -5 z^2 a^{-6} -2 z^2+a^3 z+a z+3 z a^{-1} +4 z a^{-3} +z a^{-5} +a^4+a^2+ a^{-2} + a^{-4} +1
The A2 invariant Data:K11a13/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a13/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: (-3, 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
-12 0 72 66 14 0 -32 -64 32 -288 0 -792 -168 -\frac{6511}{10} -\frac{3974}{15} -\frac{782}{15} \frac{79}{6} \frac{49}{10}

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 K11a13. 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         21 -1
9        42  2
7       42   -2
5      54    1
3     44     0
1    45      -1
-1   35       2
-3  13        -2
-5 13         2
-7 1          -1
-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}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
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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