K11a271

From Knot Atlas
Jump to: navigation, search

K11a270.gif

K11a270

K11a272.gif

K11a272

Contents

K11a271.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a271 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X12,6,13,5 X22,8,1,7 X16,10,17,9 X18,11,19,12 X8,14,9,13 X20,16,21,15 X4,18,5,17 X2,19,3,20 X14,22,15,21
Gauss code 1, -10, 2, -9, 3, -1, 4, -7, 5, -2, 6, -3, 7, -11, 8, -5, 9, -6, 10, -8, 11, -4
Dowker-Thistlethwaite code 6 10 12 22 16 18 8 20 4 2 14
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11a271 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:K11a271/ThurstonBennequinNumber
Hyperbolic Volume 18.789
A-Polynomial See Data:K11a271/A-polynomial

[edit Notes for K11a271's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11a271's four dimensional invariants]

Polynomial invariants

Alexander polynomial 3 t^3-17 t^2+40 t-51+40 t^{-1} -17 t^{-2} +3 t^{-3}
Conway polynomial 3 z^6+z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 171, 2 }
Jones polynomial q^9-4 q^8+10 q^7-17 q^6+23 q^5-28 q^4+28 q^3-24 q^2+19 q-11+5 q^{-1} - q^{-2}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +2 z^6 a^{-4} +5 z^4 a^{-4} -3 z^4 a^{-6} -z^4-2 z^2 a^{-2} +5 z^2 a^{-4} -5 z^2 a^{-6} +z^2 a^{-8} + a^{-4} -2 a^{-6} + a^{-8} +1
Kauffman polynomial (db, data sources) 4 z^{10} a^{-4} +4 z^{10} a^{-6} +12 z^9 a^{-3} +21 z^9 a^{-5} +9 z^9 a^{-7} +15 z^8 a^{-2} +18 z^8 a^{-4} +11 z^8 a^{-6} +8 z^8 a^{-8} +11 z^7 a^{-1} -13 z^7 a^{-3} -44 z^7 a^{-5} -16 z^7 a^{-7} +4 z^7 a^{-9} -23 z^6 a^{-2} -54 z^6 a^{-4} -44 z^6 a^{-6} -17 z^6 a^{-8} +z^6 a^{-10} +5 z^6+a z^5-13 z^5 a^{-1} +z^5 a^{-3} +30 z^5 a^{-5} +7 z^5 a^{-7} -8 z^5 a^{-9} +12 z^4 a^{-2} +46 z^4 a^{-4} +44 z^4 a^{-6} +12 z^4 a^{-8} -2 z^4 a^{-10} -4 z^4+2 z^3 a^{-1} -2 z^3 a^{-3} -9 z^3 a^{-5} -z^3 a^{-7} +4 z^3 a^{-9} -5 z^2 a^{-2} -15 z^2 a^{-4} -17 z^2 a^{-6} -6 z^2 a^{-8} +z^2 a^{-10} +2 z a^{-3} +4 z a^{-5} +z a^{-7} -z a^{-9} + a^{-4} +2 a^{-6} + a^{-8} +1
The A2 invariant Data:K11a271/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a271/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: (-1, -3)
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 -24 8 -\frac{206}{3} -\frac{10}{3} 96 -16 32 8 -\frac{32}{3} 288 \frac{824}{3} \frac{40}{3} \frac{23009}{30} -\frac{538}{15} \frac{23578}{45} -\frac{737}{18} \frac{2369}{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=2 is the signature of K11a271. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-1012345678χ
19           11
17          3 -3
15         71 6
13        103  -7
11       137   6
9      1510    -5
7     1313     0
5    1115      4
3   813       -5
1  412        8
-1 17         -6
-3 4          4
-51           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{4}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{15}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{15} {\mathbb Z}^{15}
r=4 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=6 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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.

K11a270.gif

K11a270

K11a272.gif

K11a272