From Knot Atlas
Jump to: navigation, search






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

Visit K11a209 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X12,3,13,4 X16,6,17,5 X20,7,21,8 X14,10,15,9 X18,11,19,12 X2,13,3,14 X8,16,9,15 X22,18,1,17 X10,19,11,20 X6,21,7,22
Gauss code 1, -7, 2, -1, 3, -11, 4, -8, 5, -10, 6, -2, 7, -5, 8, -3, 9, -6, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 12 16 20 14 18 2 8 22 10 6
A Braid Representative
A Morse Link Presentation K11a209 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a209/ThurstonBennequinNumber
Hyperbolic Volume 17.3023
A-Polynomial See Data:K11a209/A-polynomial

[edit Notes for K11a209's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11a209's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+10 t^2-34 t+51-34 t^{-1} +10 t^{-2} - t^{-3}
Conway polynomial -z^6+4 z^4-3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 141, 0 }
Jones polynomial -q^5+4 q^4-9 q^3+16 q^2-20 q+23-23 q^{-1} +19 q^{-2} -14 q^{-3} +8 q^{-4} -3 q^{-5} + q^{-6}
HOMFLY-PT polynomial (db, data sources) a^6-3 z^2 a^4-a^4+3 z^4 a^2+2 z^2 a^2-z^6-z^4-2 z^2+2 z^4 a^{-2} +z^2 a^{-2} + a^{-2} -z^2 a^{-4}
Kauffman polynomial (db, data sources) 2 a^2 z^{10}+2 z^{10}+5 a^3 z^9+12 a z^9+7 z^9 a^{-1} +5 a^4 z^8+10 a^2 z^8+10 z^8 a^{-2} +15 z^8+3 a^5 z^7-5 a^3 z^7-18 a z^7-2 z^7 a^{-1} +8 z^7 a^{-3} +a^6 z^6-9 a^4 z^6-27 a^2 z^6-14 z^6 a^{-2} +4 z^6 a^{-4} -35 z^6-7 a^5 z^5-2 a^3 z^5+9 a z^5-8 z^5 a^{-1} -11 z^5 a^{-3} +z^5 a^{-5} -3 a^6 z^4+3 a^4 z^4+20 a^2 z^4+7 z^4 a^{-2} -5 z^4 a^{-4} +26 z^4+5 a^5 z^3-a^3 z^3-10 a z^3+2 z^3 a^{-1} +5 z^3 a^{-3} -z^3 a^{-5} +3 a^6 z^2+2 a^4 z^2-7 a^2 z^2-z^2 a^{-2} +2 z^2 a^{-4} -9 z^2-a^5 z+3 a^3 z+6 a z+2 z a^{-1} -a^6-a^4- a^{-2}
The A2 invariant Data:K11a209/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a209/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: (-3, 3)
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
-12 24 72 34 6 -288 -368 -128 -8 -288 288 -408 -72 \frac{4049}{10} -\frac{1534}{15} \frac{5018}{15} -\frac{49}{6} \frac{529}{10}

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=0 is the signature of K11a209. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11           1-1
9          3 3
7         61 -5
5        103  7
3       106   -4
1      1310    3
-1     1111     0
-3    812      -4
-5   611       5
-7  28        -6
-9 16         5
-11 2          -2
-131           1
Integral Khovanov Homology

(db, data source)

\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-6 {\mathbb Z}
r=-5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-4 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=-2 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=4 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=5 {\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.