K11a293

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K11a292.gif

K11a292

K11a294.gif

K11a294

Contents

K11a293.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a293 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X16,5,17,6 X22,8,1,7 X4,10,5,9 X18,11,19,12 X20,13,21,14 X8,15,9,16 X2,17,3,18 X12,19,13,20 X14,21,15,22
Gauss code 1, -9, 2, -5, 3, -1, 4, -8, 5, -2, 6, -10, 7, -11, 8, -3, 9, -6, 10, -7, 11, -4
Dowker-Thistlethwaite code 6 10 16 22 4 18 20 8 2 12 14
A Braid Representative
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A Morse Link Presentation K11a293 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a293/ThurstonBennequinNumber
Hyperbolic Volume 13.4596
A-Polynomial See Data:K11a293/A-polynomial

[edit Notes for K11a293's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a293's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+12 t^2-15 t+15-15 t^{-1} +12 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+2 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{3,t+1\}
Determinant and Signature { 81, -4 }
Jones polynomial q^2-3 q+5-8 q^{-1} +11 q^{-2} -11 q^{-3} +13 q^{-4} -11 q^{-5} +8 q^{-6} -6 q^{-7} +3 q^{-8} - q^{-9}
HOMFLY-PT polynomial (db, data sources) a^4 z^8-a^6 z^6+6 a^4 z^6-2 a^2 z^6-4 a^6 z^4+14 a^4 z^4-9 a^2 z^4+z^4-5 a^6 z^2+17 a^4 z^2-11 a^2 z^2+3 z^2-4 a^6+8 a^4-4 a^2+1
Kauffman polynomial (db, data sources) z^3 a^{11}+3 z^4 a^{10}+6 z^5 a^9-5 z^3 a^9+2 z a^9+8 z^6 a^8-10 z^4 a^8+2 z^2 a^8+9 z^7 a^7-16 z^5 a^7+4 z^3 a^7-2 z a^7+9 z^8 a^6-24 z^6 a^6+18 z^4 a^6-11 z^2 a^6+4 a^6+6 z^9 a^5-15 z^7 a^5+z^5 a^5+10 z^3 a^5-6 z a^5+2 z^{10} a^4+3 z^8 a^4-36 z^6 a^4+51 z^4 a^4-25 z^2 a^4+8 a^4+9 z^9 a^3-40 z^7 a^3+50 z^5 a^3-15 z^3 a^3-2 z a^3+2 z^{10} a^2-5 z^8 a^2-9 z^6 a^2+28 z^4 a^2-17 z^2 a^2+4 a^2+3 z^9 a-16 z^7 a+27 z^5 a-15 z^3 a+z^8-5 z^6+8 z^4-5 z^2+1
The A2 invariant -q^{26}+q^{24}-2 q^{22}-q^{20}-q^{18}-q^{16}+4 q^{14}+4 q^{10}+q^8+q^4-2 q^2- q^{-2} + q^{-6}
The G2 invariant Data:K11a293/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: (4, -8)
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
16 -64 128 \frac{920}{3} \frac{136}{3} -1024 -\frac{5440}{3} -\frac{544}{3} -416 \frac{2048}{3} 2048 \frac{14720}{3} \frac{2176}{3} \frac{152942}{15} -\frac{3976}{5} \frac{226328}{45} \frac{1810}{9} \frac{10382}{15}

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=-4 is the signature of K11a293. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-101234χ
5           11
3          2 -2
1         31 2
-1        52  -3
-3       63   3
-5      66    0
-7     75     2
-9    46      2
-11   47       -3
-13  24        2
-15 14         -3
-17 2          2
-191           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-5 i=-3
r=-7 {\mathbb Z}
r=-6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-4 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{6}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=4 {\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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K11a292.gif

K11a292

K11a294.gif

K11a294