K11a268

From Knot Atlas
Jump to: navigation, search

K11a267.gif

K11a267

K11a269.gif

K11a269

Contents

K11a268.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a268 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X10,3,11,4 X12,6,13,5 X18,8,19,7 X20,10,21,9 X16,11,17,12 X22,13,1,14 X4,16,5,15 X2,17,3,18 X8,20,9,19 X14,21,15,22
Gauss code 1, -9, 2, -8, 3, -1, 4, -10, 5, -2, 6, -3, 7, -11, 8, -6, 9, -4, 10, -5, 11, -7
Dowker-Thistlethwaite code 6 10 12 18 20 16 22 4 2 8 14
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11a268 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a268/ThurstonBennequinNumber
Hyperbolic Volume 17.3382
A-Polynomial See Data:K11a268/A-polynomial

[edit Notes for K11a268's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a268's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-17 t^2+29 t-33+29 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6-z^4-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 139, 2 }
Jones polynomial q^7-4 q^6+9 q^5-15 q^4+20 q^3-22 q^2+22 q-19+14 q^{-1} -8 q^{-2} +4 q^{-3} - q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -5 z^6 a^{-2} +z^6 a^{-4} +2 z^6-a^2 z^4-10 z^4 a^{-2} +3 z^4 a^{-4} +7 z^4-2 a^2 z^2-9 z^2 a^{-2} +3 z^2 a^{-4} +7 z^2-2 a^{-2} + a^{-4} +2
Kauffman polynomial (db, data sources) 3 z^{10} a^{-2} +3 z^{10}+6 a z^9+16 z^9 a^{-1} +10 z^9 a^{-3} +4 a^2 z^8+15 z^8 a^{-2} +15 z^8 a^{-4} +4 z^8+a^3 z^7-19 a z^7-44 z^7 a^{-1} -10 z^7 a^{-3} +14 z^7 a^{-5} -14 a^2 z^6-57 z^6 a^{-2} -25 z^6 a^{-4} +9 z^6 a^{-6} -37 z^6-3 a^3 z^5+17 a z^5+32 z^5 a^{-1} -11 z^5 a^{-3} -19 z^5 a^{-5} +4 z^5 a^{-7} +15 a^2 z^4+51 z^4 a^{-2} +13 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +45 z^4+2 a^3 z^3-4 a z^3-7 z^3 a^{-1} +9 z^3 a^{-3} +9 z^3 a^{-5} -z^3 a^{-7} -5 a^2 z^2-18 z^2 a^{-2} -4 z^2 a^{-4} +2 z^2 a^{-6} -17 z^2-z a^{-3} -z a^{-5} +2 a^{-2} + a^{-4} +2
The A2 invariant -q^{12}+q^{10}+q^8-q^6+4 q^4-3 q^2+1+ q^{-2} -3 q^{-4} +5 q^{-6} -4 q^{-8} +3 q^{-10} - q^{-12} -2 q^{-14} +3 q^{-16} -2 q^{-18} + q^{-20}
The G2 invariant Data:K11a268/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{34}{3} \frac{38}{3} 32 \frac{208}{3} \frac{256}{3} -40 -\frac{32}{3} 32 -\frac{136}{3} -\frac{152}{3} \frac{2129}{30} \frac{3062}{15} -\frac{10502}{45} \frac{1039}{18} -\frac{1711}{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 K11a268. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
15           11
13          3 -3
11         61 5
9        93  -6
7       116   5
5      119    -2
3     1111     0
1    912      3
-1   510       -5
-3  39        6
-5 15         -4
-7 3          3
-91           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{11}
r=1 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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.

K11a267.gif

K11a267

K11a269.gif

K11a269