K11a105

From Knot Atlas
Jump to: navigation, search

K11a104.gif

K11a104

K11a106.gif

K11a106

Contents

K11a105.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a105 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,6,15,5 X16,8,17,7 X2,10,3,9 X20,11,21,12 X8,14,9,13 X6,16,7,15 X22,18,1,17 X12,19,13,20 X18,22,19,21
Gauss code 1, -5, 2, -1, 3, -8, 4, -7, 5, -2, 6, -10, 7, -3, 8, -4, 9, -11, 10, -6, 11, -9
Dowker-Thistlethwaite code 4 10 14 16 2 20 8 6 22 12 18
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11a105 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a105/ThurstonBennequinNumber
Hyperbolic Volume 14.717
A-Polynomial See Data:K11a105/A-polynomial

[edit Notes for K11a105's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a105's four dimensional invariants]

Polynomial invariants

Alexander polynomial -3 t^3+13 t^2-24 t+29-24 t^{-1} +13 t^{-2} -3 t^{-3}
Conway polynomial -3 z^6-5 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 109, 4 }
Jones polynomial -q^{11}+4 q^{10}-8 q^9+12 q^8-16 q^7+18 q^6-17 q^5+14 q^4-10 q^3+6 q^2-2 q+1
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-4} -2 z^6 a^{-6} +z^4 a^{-2} -2 z^4 a^{-4} -7 z^4 a^{-6} +3 z^4 a^{-8} +3 z^2 a^{-2} -8 z^2 a^{-6} +7 z^2 a^{-8} -z^2 a^{-10} +2 a^{-2} -3 a^{-6} +3 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +3 z^9 a^{-5} +7 z^9 a^{-7} +4 z^9 a^{-9} +3 z^8 a^{-4} +7 z^8 a^{-6} +11 z^8 a^{-8} +7 z^8 a^{-10} +2 z^7 a^{-3} -5 z^7 a^{-5} -12 z^7 a^{-7} +2 z^7 a^{-9} +7 z^7 a^{-11} +z^6 a^{-2} -6 z^6 a^{-4} -23 z^6 a^{-6} -30 z^6 a^{-8} -10 z^6 a^{-10} +4 z^6 a^{-12} -5 z^5 a^{-3} +5 z^5 a^{-5} +9 z^5 a^{-7} -14 z^5 a^{-9} -12 z^5 a^{-11} +z^5 a^{-13} -4 z^4 a^{-2} +2 z^4 a^{-4} +29 z^4 a^{-6} +33 z^4 a^{-8} +4 z^4 a^{-10} -6 z^4 a^{-12} +2 z^3 a^{-3} -7 z^3 a^{-5} -4 z^3 a^{-7} +11 z^3 a^{-9} +5 z^3 a^{-11} -z^3 a^{-13} +5 z^2 a^{-2} -17 z^2 a^{-6} -15 z^2 a^{-8} -2 z^2 a^{-10} +z^2 a^{-12} +z a^{-3} +2 z a^{-5} +z a^{-7} -z a^{-9} -z a^{-11} -2 a^{-2} +3 a^{-6} +3 a^{-8} + a^{-10}
The A2 invariant Data:K11a105/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a105/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, 2)
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 16 8 \frac{350}{3} \frac{130}{3} 64 \frac{1792}{3} \frac{352}{3} 208 \frac{32}{3} 128 \frac{1400}{3} \frac{520}{3} \frac{72271}{30} -\frac{2554}{5} \frac{83102}{45} \frac{3089}{18} \frac{6511}{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 K11a105. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
23           1-1
21          3 3
19         51 -4
17        73  4
15       95   -4
13      97    2
11     89     1
9    69      -3
7   48       4
5  26        -4
3 15         4
1 1          -1
-11           1
Integral Khovanov Homology

(db, data source)

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

K11a104.gif

K11a104

K11a106.gif

K11a106