K11a4

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K11a3.gif

K11a3

K11a5.gif

K11a5

Contents

K11a4.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a4 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X18,11,19,12 X6,14,7,13 X22,15,1,16 X20,17,21,18 X12,19,13,20 X16,21,17,22
Gauss code 1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -10, 7, -4, 8, -11, 9, -6, 10, -9, 11, -8
Dowker-Thistlethwaite code 4 8 10 14 2 18 6 22 20 12 16
A Braid Representative
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A Morse Link Presentation K11a4 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:K11a4/ThurstonBennequinNumber
Hyperbolic Volume 14.1846
A-Polynomial See Data:K11a4/A-polynomial

[edit Notes for K11a4'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 K11a4's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+10 t^2-22 t+29-22 t^{-1} +10 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-2 z^4+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 97, 0 }
Jones polynomial q^4-3 q^3+6 q^2-10 q+14-15 q^{-1} +15 q^{-2} -13 q^{-3} +10 q^{-4} -6 q^{-5} +3 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6-a^6+2 z^4 a^4+4 z^2 a^4+2 a^4-z^6 a^2-2 z^4 a^2-z^2 a^2-z^6-3 z^4-4 z^2-1+z^4 a^{-2} +2 z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) a^4 z^{10}+a^2 z^{10}+3 a^5 z^9+7 a^3 z^9+4 a z^9+3 a^6 z^8+7 a^4 z^8+10 a^2 z^8+6 z^8+a^7 z^7-7 a^5 z^7-15 a^3 z^7-a z^7+6 z^7 a^{-1} -12 a^6 z^6-34 a^4 z^6-33 a^2 z^6+5 z^6 a^{-2} -6 z^6-4 a^7 z^5-3 a^5 z^5-a^3 z^5-10 a z^5-5 z^5 a^{-1} +3 z^5 a^{-3} +15 a^6 z^4+39 a^4 z^4+28 a^2 z^4-5 z^4 a^{-2} +z^4 a^{-4} -2 z^4+5 a^7 z^3+11 a^5 z^3+10 a^3 z^3+7 a z^3-3 z^3 a^{-3} -7 a^6 z^2-16 a^4 z^2-8 a^2 z^2+3 z^2 a^{-2} -z^2 a^{-4} +5 z^2-2 a^7 z-4 a^5 z-3 a^3 z-a z+z a^{-1} +z a^{-3} +a^6+2 a^4- a^{-2} -1
The A2 invariant Data:K11a4/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a4/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a110,}

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

Vassiliev invariants

V2 and V3: (0, -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
0 -16 0 48 16 0 -\frac{64}{3} \frac{32}{3} 16 0 128 0 0 216 \frac{352}{3} -\frac{352}{3} \frac{104}{3} -40

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 K11a4. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-7-6-5-4-3-2-101234χ
9           11
7          2 -2
5         41 3
3        62  -4
1       84   4
-1      87    -1
-3     77     0
-5    68      2
-7   47       -3
-9  26        4
-11 14         -3
-13 2          2
-151           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
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}^{6}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
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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K11a3.gif

K11a3

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K11a5