From Knot Atlas
Jump to: navigation, search






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

Visit K11n131 at Knotilus!

Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X18,5,19,6 X7,16,8,17 X9,14,10,15 X2,11,3,12 X20,14,21,13 X15,8,16,9 X22,17,1,18 X12,20,13,19 X6,21,7,22
Gauss code 1, -6, 2, -1, 3, -11, -4, 8, -5, -2, 6, -10, 7, 5, -8, 4, 9, -3, 10, -7, 11, -9
Dowker-Thistlethwaite code 4 10 18 -16 -14 2 20 -8 22 12 6
A Braid Representative
A Morse Link Presentation K11n131 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:K11n131/ThurstonBennequinNumber
Hyperbolic Volume 14.3723
A-Polynomial See Data:K11n131/A-polynomial

[edit Notes for K11n131's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 2

[edit Notes for K11n131's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-6 t^2+16 t-21+16 t^{-1} -6 t^{-2} + t^{-3}
Conway polynomial z^6+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 67, -2 }
Jones polynomial -q^2+4 q-7+10 q^{-1} -11 q^{-2} +12 q^{-3} -10 q^{-4} +7 q^{-5} -4 q^{-6} + q^{-7}
HOMFLY-PT polynomial (db, data sources) z^2 a^6-2 z^4 a^4-3 z^2 a^4-a^4+z^6 a^2+3 z^4 a^2+4 z^2 a^2+2 a^2-z^4-z^2
Kauffman polynomial (db, data sources) a^5 z^9+a^3 z^9+a^6 z^8+5 a^4 z^8+4 a^2 z^8+a^5 z^7+7 a^3 z^7+6 a z^7+a^6 z^6-5 a^4 z^6-2 a^2 z^6+4 z^6+4 a^7 z^5+2 a^5 z^5-14 a^3 z^5-11 a z^5+z^5 a^{-1} +a^8 z^4+a^6 z^4-a^4 z^4-8 a^2 z^4-7 z^4-4 a^7 z^3-3 a^5 z^3+5 a^3 z^3+3 a z^3-z^3 a^{-1} -a^8 z^2-a^6 z^2+4 a^4 z^2+6 a^2 z^2+2 z^2+a^5 z+a^3 z-a^4-2 a^2
The A2 invariant Data:K11n131/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n131/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n7, K11n160,}

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{62}{3} \frac{10}{3} -32 -\frac{176}{3} -\frac{128}{3} 24 \frac{32}{3} 32 \frac{248}{3} \frac{40}{3} \frac{5071}{30} \frac{2938}{15} -\frac{2698}{45} -\frac{1135}{18} -\frac{689}{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 K11n131. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
5         1-1
3        3 3
1       41 -3
-1      63  3
-3     65   -1
-5    65    1
-7   46     2
-9  36      -3
-11 14       3
-13 3        -3
-151         1
Integral Khovanov Homology

(db, data source)

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