K11a252

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K11a251.gif

K11a251

K11a253.gif

K11a253

Contents

K11a252.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a252 at Knotilus!



Knot presentations

Planar diagram presentation X6271 X8493 X12,5,13,6 X2837 X18,9,19,10 X16,12,17,11 X4,13,5,14 X20,16,21,15 X22,17,1,18 X14,20,15,19 X10,21,11,22
Gauss code 1, -4, 2, -7, 3, -1, 4, -2, 5, -11, 6, -3, 7, -10, 8, -6, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code 6 8 12 2 18 16 4 20 22 14 10
A Braid Representative
BraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart3.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart4.gif
A Morse Link Presentation K11a252 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:K11a252/ThurstonBennequinNumber
Hyperbolic Volume 16.8559
A-Polynomial See Data:K11a252/A-polynomial

[edit Notes for K11a252'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 K11a252's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-16 t^2+27 t-31+27 t^{-1} -16 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 131, 2 }
Jones polynomial q^7-4 q^6+9 q^5-15 q^4+19 q^3-21 q^2+21 q-17+13 q^{-1} -7 q^{-2} +3 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} +8 z^4-3 a^2 z^2-10 z^2 a^{-2} +3 z^2 a^{-4} +11 z^2-2 a^2-4 a^{-2} + a^{-4} +6
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10}+4 a z^9+12 z^9 a^{-1} +8 z^9 a^{-3} +3 a^2 z^8+17 z^8 a^{-2} +14 z^8 a^{-4} +6 z^8+a^3 z^7-10 a z^7-26 z^7 a^{-1} -z^7 a^{-3} +14 z^7 a^{-5} -11 a^2 z^6-55 z^6 a^{-2} -22 z^6 a^{-4} +9 z^6 a^{-6} -35 z^6-4 a^3 z^5+2 a z^5+3 z^5 a^{-1} -27 z^5 a^{-3} -20 z^5 a^{-5} +4 z^5 a^{-7} +14 a^2 z^4+47 z^4 a^{-2} +10 z^4 a^{-4} -7 z^4 a^{-6} +z^4 a^{-8} +43 z^4+5 a^3 z^3+9 a z^3+15 z^3 a^{-1} +23 z^3 a^{-3} +11 z^3 a^{-5} -z^3 a^{-7} -8 a^2 z^2-19 z^2 a^{-2} -3 z^2 a^{-4} +2 z^2 a^{-6} -22 z^2-2 a^3 z-5 a z-7 z a^{-1} -6 z a^{-3} -2 z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6
The A2 invariant -q^{12}-2 q^6+4 q^4-q^2+3+3 q^{-2} -3 q^{-4} +4 q^{-6} -5 q^{-8} +2 q^{-10} - q^{-12} -2 q^{-14} +3 q^{-16} -2 q^{-18} + q^{-20}
The G2 invariant Data:K11a252/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a131, K11a254,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a254,}

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{14}{3} \frac{10}{3} -32 -\frac{80}{3} \frac{160}{3} -40 \frac{32}{3} 32 \frac{56}{3} \frac{40}{3} \frac{2911}{30} \frac{1658}{15} -\frac{3418}{45} \frac{737}{18} -\frac{449}{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 K11a252. 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       106   4
5      119    -2
3     1010     0
1    812      4
-1   59       -4
-3  28        6
-5 15         -4
-7 2          2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
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).

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K11a251.gif

K11a251

K11a253.gif

K11a253