K11a311

From Knot Atlas
Jump to: navigation, search

K11a310.gif

K11a310

K11a312.gif

K11a312

Contents

K11a311.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a311 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X12,4,13,3 X14,6,15,5 X20,8,21,7 X18,10,19,9 X4,12,5,11 X2,14,3,13 X22,15,1,16 X10,18,11,17 X8,20,9,19 X16,21,17,22
Gauss code 1, -7, 2, -6, 3, -1, 4, -10, 5, -9, 6, -2, 7, -3, 8, -11, 9, -5, 10, -4, 11, -8
Dowker-Thistlethwaite code 6 12 14 20 18 4 2 22 10 8 16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11a311 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 2
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a311/ThurstonBennequinNumber
Hyperbolic Volume 11.8941
A-Polynomial See Data:K11a311/A-polynomial

[edit Notes for K11a311's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 2
Rasmussen s-Invariant -2

[edit Notes for K11a311's four dimensional invariants]

Polynomial invariants

Alexander polynomial -4 t^2+20 t-31+20 t^{-1} -4 t^{-2}
Conway polynomial -4 z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 79, 2 }
Jones polynomial -q^{10}+2 q^9-4 q^8+7 q^7-10 q^6+12 q^5-12 q^4+12 q^3-9 q^2+6 q-3+ q^{-1}
HOMFLY-PT polynomial (db, data sources) -z^4 a^{-2} -2 z^4 a^{-4} -z^4 a^{-6} +z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} +2 z^2 a^{-8} +z^2+ a^{-2} + a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +3 z^9 a^{-5} +5 z^9 a^{-7} +2 z^9 a^{-9} +5 z^8 a^{-4} +3 z^8 a^{-6} +2 z^8 a^{-10} +6 z^7 a^{-3} -3 z^7 a^{-5} -17 z^7 a^{-7} -7 z^7 a^{-9} +z^7 a^{-11} +5 z^6 a^{-2} -7 z^6 a^{-4} -11 z^6 a^{-6} -8 z^6 a^{-8} -9 z^6 a^{-10} +3 z^5 a^{-1} -9 z^5 a^{-3} +z^5 a^{-5} +25 z^5 a^{-7} +7 z^5 a^{-9} -5 z^5 a^{-11} -6 z^4 a^{-2} +3 z^4 a^{-4} +9 z^4 a^{-6} +11 z^4 a^{-8} +12 z^4 a^{-10} +z^4-3 z^3 a^{-1} +6 z^3 a^{-3} -3 z^3 a^{-5} -23 z^3 a^{-7} -4 z^3 a^{-9} +7 z^3 a^{-11} +3 z^2 a^{-2} +2 z^2 a^{-4} -4 z^2 a^{-6} -7 z^2 a^{-8} -5 z^2 a^{-10} -z^2+2 z a^{-5} +6 z a^{-7} +2 z a^{-9} -2 z a^{-11} - a^{-2} + a^{-8} + a^{-10}
The A2 invariant Data:K11a311/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a311/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, 10)
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 80 128 \frac{1400}{3} \frac{280}{3} 1280 \frac{8768}{3} \frac{1280}{3} 656 \frac{2048}{3} 3200 \frac{22400}{3} \frac{4480}{3} \frac{272702}{15} -\frac{31048}{15} \frac{463208}{45} \frac{3970}{9} \frac{23582}{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=2 is the signature of K11a311. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
21           1-1
19          1 1
17         31 -2
15        41  3
13       63   -3
11      64    2
9     66     0
7    66      0
5   36       3
3  36        -3
1 14         3
-1 2          -2
-31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=9 {\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.

K11a310.gif

K11a310

K11a312.gif

K11a312