K11a250

From Knot Atlas
Jump to: navigation, search

K11a249.gif

K11a249

K11a251.gif

K11a251

Contents

K11a250.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a250 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X8493 X12,5,13,6 X2837 X14,10,15,9 X18,11,19,12 X4,13,5,14 X20,16,21,15 X22,18,1,17 X10,19,11,20 X16,22,17,21
Gauss code 1, -4, 2, -7, 3, -1, 4, -2, 5, -10, 6, -3, 7, -5, 8, -11, 9, -6, 10, -8, 11, -9
Dowker-Thistlethwaite code 6 8 12 2 14 18 4 20 22 10 16
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a250 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a250/ThurstonBennequinNumber
Hyperbolic Volume 13.803
A-Polynomial See Data:K11a250/A-polynomial

[edit Notes for K11a250's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a250's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+11 t^2-16 t+19-16 t^{-1} +11 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 85, 4 }
Jones polynomial -q^9+3 q^8-6 q^7+10 q^6-12 q^5+13 q^4-13 q^3+11 q^2-8 q+5-2 q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +14 z^4 a^{-4} -4 z^4 a^{-6} +z^4-16 z^2 a^{-2} +16 z^2 a^{-4} -5 z^2 a^{-6} +4 z^2-8 a^{-2} +7 a^{-4} -2 a^{-6} +4
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +7 z^9 a^{-3} +5 z^9 a^{-5} +2 z^8 a^{-2} +11 z^8 a^{-4} +10 z^8 a^{-6} +z^8-10 z^7 a^{-1} -26 z^7 a^{-3} -4 z^7 a^{-5} +12 z^7 a^{-7} -27 z^6 a^{-2} -54 z^6 a^{-4} -23 z^6 a^{-6} +10 z^6 a^{-8} -6 z^6+16 z^5 a^{-1} +22 z^5 a^{-3} -26 z^5 a^{-5} -26 z^5 a^{-7} +6 z^5 a^{-9} +54 z^4 a^{-2} +68 z^4 a^{-4} +9 z^4 a^{-6} -15 z^4 a^{-8} +3 z^4 a^{-10} +13 z^4-8 z^3 a^{-1} +4 z^3 a^{-3} +31 z^3 a^{-5} +15 z^3 a^{-7} -3 z^3 a^{-9} +z^3 a^{-11} -37 z^2 a^{-2} -33 z^2 a^{-4} -2 z^2 a^{-6} +6 z^2 a^{-8} -12 z^2-5 z a^{-3} -9 z a^{-5} -4 z a^{-7} +8 a^{-2} +7 a^{-4} +2 a^{-6} +4
The A2 invariant q^6+q^4+q^2+2-2 q^{-2} -2 q^{-6} -2 q^{-8} +2 q^{-10} -2 q^{-12} +4 q^{-14} + q^{-18} + q^{-20} -2 q^{-22} + q^{-24} - q^{-26}
The G2 invariant Data:K11a250/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, 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{10}{3} -32 -\frac{208}{3} -\frac{352}{3} 8 -\frac{32}{3} 32 -\frac{520}{3} \frac{40}{3} -\frac{11791}{30} -\frac{8378}{15} \frac{11338}{45} -\frac{593}{18} \frac{2609}{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 K11a250. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
19           1-1
17          2 2
15         41 -3
13        62  4
11       64   -2
9      76    1
7     66     0
5    57      -2
3   47       3
1  14        -3
-1 14         3
-3 1          -1
-51           1
Integral Khovanov Homology

(db, data source)

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

K11a249.gif

K11a249

K11a251.gif

K11a251