K11n135

From Knot Atlas
Jump to: navigation, search

K11n134.gif

K11n134

K11n136.gif

K11n136

Contents

K11n135.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n135 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X14,6,15,5 X7,17,8,16 X22,10,1,9 X11,19,12,18 X2,13,3,14 X15,20,16,21 X17,11,18,10 X19,7,20,6 X8,22,9,21
Gauss code 1, -7, 2, -1, 3, 10, -4, -11, 5, 9, -6, -2, 7, -3, -8, 4, -9, 6, -10, 8, 11, -5
Dowker-Thistlethwaite code 4 12 14 -16 22 -18 2 -20 -10 -6 8
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11n135 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n135/ThurstonBennequinNumber
Hyperbolic Volume 8.03988
A-Polynomial See Data:K11n135/A-polynomial

[edit Notes for K11n135's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -4

[edit Notes for K11n135's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+2 t^2-1+2 t^{-2} - t^{-3}
Conway polynomial -z^6-4 z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 5, 4 }
Jones polynomial q^9-q^8-q^5+q^4-q^3+2 q^2-q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} +z^4 a^{-2} -5 z^4 a^{-4} +4 z^2 a^{-2} -6 z^2 a^{-4} +z^2 a^{-8} +3 a^{-2} -2 a^{-4} - a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) z^7 a^{-3} +z^7 a^{-9} +z^6 a^{-2} +2 z^6 a^{-4} +z^6 a^{-10} -4 z^5 a^{-3} +z^5 a^{-5} -z^5 a^{-7} -6 z^5 a^{-9} -5 z^4 a^{-2} -9 z^4 a^{-4} -z^4 a^{-8} -5 z^4 a^{-10} +2 z^3 a^{-3} -4 z^3 a^{-5} +3 z^3 a^{-7} +9 z^3 a^{-9} +7 z^2 a^{-2} +9 z^2 a^{-4} -2 z^2 a^{-6} +z^2 a^{-8} +5 z^2 a^{-10} +2 z a^{-3} +2 z a^{-5} -3 z a^{-7} -3 z a^{-9} -3 a^{-2} -2 a^{-4} + a^{-6} + a^{-8}
The A2 invariant 1+ q^{-2} + q^{-4} + q^{-6} + q^{-8} + q^{-10} - q^{-12} -2 q^{-16} - q^{-18} - q^{-20} + q^{-24} + q^{-28}
The G2 invariant Data:K11n135/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n19,}

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

Vassiliev invariants

V2 and V3: (-1, -4)
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 -32 8 -\frac{158}{3} \frac{110}{3} 128 \frac{736}{3} \frac{352}{3} 192 -\frac{32}{3} 512 \frac{632}{3} -\frac{440}{3} \frac{57329}{30} -\frac{806}{5} \frac{55378}{45} \frac{4495}{18} \frac{1649}{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=4 is the signature of K11n135. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-1012345678χ
19          11
17           0
15       111 -1
13      11   0
11     111   -1
9    121    0
7   11      0
5  111      1
3 12        1
1           0
-11          1
Integral Khovanov Homology

(db, data source)

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

Back to the top.

K11n134.gif

K11n134

K11n136.gif

K11n136