K11a313

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K11a312.gif

K11a312

K11a314.gif

K11a314

Contents

K11a313.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a313 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,3,13,4 X16,5,17,6 X14,8,15,7 X20,9,21,10 X18,11,19,12 X2,13,3,14 X22,16,1,15 X4,17,5,18 X10,19,11,20 X8,21,9,22
Gauss code 1, -7, 2, -9, 3, -1, 4, -11, 5, -10, 6, -2, 7, -4, 8, -3, 9, -6, 10, -5, 11, -8
Dowker-Thistlethwaite code 6 12 16 14 20 18 2 22 4 10 8
A Braid Representative
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A Morse Link Presentation K11a313 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a313/ThurstonBennequinNumber
Hyperbolic Volume 13.3553
A-Polynomial See Data:K11a313/A-polynomial

[edit Notes for K11a313'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 K11a313's four dimensional invariants]

Polynomial invariants

Alexander polynomial 5 t^2-19 t+29-19 t^{-1} +5 t^{-2}
Conway polynomial 5 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 77, 0 }
Jones polynomial -q^3+3 q^2-5 q+8-10 q^{-1} +12 q^{-2} -11 q^{-3} +10 q^{-4} -8 q^{-5} +5 q^{-6} -3 q^{-7} + q^{-8}
HOMFLY-PT polynomial (db, data sources) a^8-3 z^2 a^6-3 a^6+2 z^4 a^4+3 z^2 a^4+2 a^4+2 z^4 a^2+2 z^2 a^2+a^2+z^4-z^2 a^{-2}
Kauffman polynomial (db, data sources) 2 a^6 z^{10}+2 a^4 z^{10}+3 a^7 z^9+9 a^5 z^9+6 a^3 z^9+a^8 z^8-5 a^6 z^8+2 a^4 z^8+8 a^2 z^8-16 a^7 z^7-42 a^5 z^7-18 a^3 z^7+8 a z^7-5 a^8 z^6-10 a^6 z^6-33 a^4 z^6-21 a^2 z^6+7 z^6+27 a^7 z^5+59 a^5 z^5+12 a^3 z^5-15 a z^5+5 z^5 a^{-1} +8 a^8 z^4+31 a^6 z^4+47 a^4 z^4+12 a^2 z^4+3 z^4 a^{-2} -9 z^4-16 a^7 z^3-29 a^5 z^3-5 a^3 z^3+4 a z^3-3 z^3 a^{-1} +z^3 a^{-3} -5 a^8 z^2-19 a^6 z^2-19 a^4 z^2-2 a^2 z^2-z^2 a^{-2} +2 z^2+2 a^7 z+5 a^5 z+3 a^3 z+a^8+3 a^6+2 a^4-a^2
The A2 invariant Data:K11a313/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a313/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{178}{3} -\frac{110}{3} 0 320 -32 256 \frac{32}{3} 0 -\frac{712}{3} -\frac{440}{3} -\frac{34049}{30} \frac{18898}{15} -\frac{83458}{45} -\frac{4639}{18} -\frac{11489}{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 K11a313. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-8-7-6-5-4-3-2-10123χ
7           1-1
5          2 2
3         31 -2
1        52  3
-1       64   -2
-3      64    2
-5     56     1
-7    56      -1
-9   35       2
-11  25        -3
-13 13         2
-15 2          -2
-171           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-8 {\mathbb Z}
r=-7 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{5}
r=1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\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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K11a312

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K11a314