K11a8

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K11a7.gif

K11a7

K11a9.gif

K11a9

Contents

K11a8.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a8 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X16,8,17,7 X2,9,3,10 X18,11,19,12 X20,13,21,14 X6,16,7,15 X22,17,1,18 X14,19,15,20 X12,21,13,22
Gauss code 1, -5, 2, -1, 3, -8, 4, -2, 5, -3, 6, -11, 7, -10, 8, -4, 9, -6, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 8 10 16 2 18 20 6 22 14 12
A Braid Representative
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A Morse Link Presentation K11a8 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a8/ThurstonBennequinNumber
Hyperbolic Volume 14.9162
A-Polynomial See Data:K11a8/A-polynomial

[edit Notes for K11a8's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus [2,3]
Rasmussen s-Invariant 0

[edit Notes for K11a8's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+11 t^2-27 t+37-27 t^{-1} +11 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 117, 0 }
Jones polynomial q^4-4 q^3+9 q^2-13 q+17-19 q^{-1} +18 q^{-2} -15 q^{-3} +11 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+3 a^4-z^6 a^2-2 z^4 a^2-3 z^2 a^2-2 a^2-z^6-2 z^4-2 z^2+z^4 a^{-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+8 a^3 z^9+5 a z^9+3 a^6 z^8+9 a^4 z^8+16 a^2 z^8+10 z^8+a^7 z^7-6 a^5 z^7-12 a^3 z^7+7 a z^7+12 z^7 a^{-1} -12 a^6 z^6-39 a^4 z^6-45 a^2 z^6+9 z^6 a^{-2} -9 z^6-4 a^7 z^5-5 a^5 z^5-13 a^3 z^5-32 a z^5-16 z^5 a^{-1} +4 z^5 a^{-3} +16 a^6 z^4+46 a^4 z^4+35 a^2 z^4-9 z^4 a^{-2} +z^4 a^{-4} -5 z^4+5 a^7 z^3+14 a^5 z^3+21 a^3 z^3+21 a z^3+8 z^3 a^{-1} -z^3 a^{-3} -8 a^6 z^2-20 a^4 z^2-13 a^2 z^2+4 z^2 a^{-2} +3 z^2-2 a^7 z-5 a^5 z-5 a^3 z-4 a z-2 z a^{-1} +a^6+3 a^4+2 a^2- a^{-2}
The A2 invariant Data:K11a8/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a8/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a38, K11a187, K11a249,}

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

Vassiliev invariants

V2 and V3: (-1, -1)
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 -8 8 \frac{130}{3} \frac{38}{3} 32 \frac{112}{3} \frac{64}{3} 24 -\frac{32}{3} 32 -\frac{520}{3} -\frac{152}{3} -\frac{6991}{30} \frac{754}{5} -\frac{17702}{45} \frac{751}{18} -\frac{2191}{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 K11a8. 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          3 -3
5         61 5
3        73  -4
1       106   4
-1      108    -2
-3     89     -1
-5    710      3
-7   48       -4
-9  27        5
-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}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a7

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K11a9