K11a200

From Knot Atlas
Jump to: navigation, search

K11a199.gif

K11a199

K11a201.gif

K11a201

Contents

K11a200.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a200 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X16,6,17,5 X14,8,15,7 X20,10,21,9 X2,12,3,11 X10,14,11,13 X6,16,7,15 X22,18,1,17 X8,20,9,19 X18,22,19,21
Gauss code 1, -6, 2, -1, 3, -8, 4, -10, 5, -7, 6, -2, 7, -4, 8, -3, 9, -11, 10, -5, 11, -9
Dowker-Thistlethwaite code 4 12 16 14 20 2 10 6 22 8 18
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart2.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart1.gifBraidPart0.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart3.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart4.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11a200 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a200/ThurstonBennequinNumber
Hyperbolic Volume 13.7591
A-Polynomial See Data:K11a200/A-polynomial

[edit Notes for K11a200's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a200's four dimensional invariants]

Polynomial invariants

Alexander polynomial 7 t^2-21 t+29-21 t^{-1} +7 t^{-2}
Conway polynomial 7 z^4+7 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 85, 4 }
Jones polynomial -q^{13}+3 q^{12}-6 q^{11}+9 q^{10}-12 q^9+13 q^8-13 q^7+12 q^6-8 q^5+5 q^4-2 q^3+q^2
HOMFLY-PT polynomial (db, data sources) z^4 a^{-4} +2 z^4 a^{-6} +3 z^4 a^{-8} +z^4 a^{-10} +2 z^2 a^{-4} +2 z^2 a^{-6} +5 z^2 a^{-8} -z^2 a^{-10} -z^2 a^{-12} + a^{-4} +2 a^{-8} -2 a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-10} +z^{10} a^{-12} +3 z^9 a^{-9} +6 z^9 a^{-11} +3 z^9 a^{-13} +5 z^8 a^{-8} +7 z^8 a^{-10} +5 z^8 a^{-12} +3 z^8 a^{-14} +4 z^7 a^{-7} -13 z^7 a^{-11} -8 z^7 a^{-13} +z^7 a^{-15} +3 z^6 a^{-6} -11 z^6 a^{-8} -28 z^6 a^{-10} -26 z^6 a^{-12} -12 z^6 a^{-14} +2 z^5 a^{-5} -4 z^5 a^{-7} -15 z^5 a^{-9} -5 z^5 a^{-11} -4 z^5 a^{-15} +z^4 a^{-4} -2 z^4 a^{-6} +15 z^4 a^{-8} +29 z^4 a^{-10} +25 z^4 a^{-12} +14 z^4 a^{-14} -2 z^3 a^{-5} +2 z^3 a^{-7} +19 z^3 a^{-9} +17 z^3 a^{-11} +7 z^3 a^{-13} +5 z^3 a^{-15} -2 z^2 a^{-4} -9 z^2 a^{-8} -12 z^2 a^{-10} -6 z^2 a^{-12} -5 z^2 a^{-14} -6 z a^{-9} -7 z a^{-11} -3 z a^{-13} -2 z a^{-15} + a^{-4} +2 a^{-8} +2 a^{-10}
The A2 invariant Data:K11a200/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a200/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_101,}

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

Vassiliev invariants

V2 and V3: (7, 19)
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
28 152 392 \frac{3026}{3} \frac{406}{3} 4256 \frac{22544}{3} \frac{3776}{3} 888 \frac{10976}{3} 11552 \frac{84728}{3} \frac{11368}{3} \frac{1723657}{30} \frac{13122}{5} \frac{899594}{45} \frac{8087}{18} \frac{73417}{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 K11a200. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
27           1-1
25          2 2
23         41 -3
21        52  3
19       74   -3
17      65    1
15     77     0
13    56      -1
11   37       4
9  25        -3
7  3         3
512          -1
31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{2}
r=2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=7 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=8 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=9 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=11 {\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.

K11a199.gif

K11a199

K11a201.gif

K11a201