From Knot Atlas
Jump to: navigation, search






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

Visit K11a256 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X8394 X12,6,13,5 X20,8,21,7 X18,9,19,10 X16,11,17,12 X22,13,1,14 X4,16,5,15 X10,17,11,18 X2,19,3,20 X14,21,15,22
Gauss code 1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, 7, -11, 8, -6, 9, -5, 10, -4, 11, -7
Dowker-Thistlethwaite code 6 8 12 20 18 16 22 4 10 2 14
A Braid Representative
A Morse Link Presentation K11a256 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:K11a256/ThurstonBennequinNumber
Hyperbolic Volume 16.7429
A-Polynomial See Data:K11a256/A-polynomial

[edit Notes for K11a256'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 K11a256's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+12 t^2-31 t+43-31 t^{-1} +12 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 133, 0 }
Jones polynomial q^4-4 q^3+9 q^2-14 q+19-21 q^{-1} +21 q^{-2} -18 q^{-3} +13 q^{-4} -8 q^{-5} +4 q^{-6} - q^{-7}
HOMFLY-PT polynomial (db, data sources) -z^2 a^6+2 z^4 a^4+2 z^2 a^4-z^6 a^2-z^4 a^2+a^2-z^6-2 z^4-3 z^2-1+z^4 a^{-2} +z^2 a^{-2} + a^{-2}
Kauffman polynomial (db, data sources) 3 a^4 z^{10}+3 a^2 z^{10}+6 a^5 z^9+15 a^3 z^9+9 a z^9+4 a^6 z^8+2 a^4 z^8+11 a^2 z^8+13 z^8+a^7 z^7-20 a^5 z^7-43 a^3 z^7-9 a z^7+13 z^7 a^{-1} -14 a^6 z^6-30 a^4 z^6-43 a^2 z^6+9 z^6 a^{-2} -18 z^6-3 a^7 z^5+19 a^5 z^5+34 a^3 z^5-9 a z^5-17 z^5 a^{-1} +4 z^5 a^{-3} +14 a^6 z^4+35 a^4 z^4+32 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +2 z^4+2 a^7 z^3-5 a^5 z^3-8 a^3 z^3+7 a z^3+7 z^3 a^{-1} -z^3 a^{-3} -4 a^6 z^2-10 a^4 z^2-5 a^2 z^2+3 z^2 a^{-2} +4 z^2-a z-z a^{-1} -a^2- a^{-2} -1
The A2 invariant Data:K11a256/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a256/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, 0)
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 0 8 \frac{34}{3} \frac{14}{3} 0 0 64 -64 -\frac{32}{3} 0 -\frac{136}{3} -\frac{56}{3} -\frac{1231}{30} -\frac{3218}{15} \frac{4018}{45} \frac{1807}{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=0 is the signature of K11a256. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
9           11
7          3 -3
5         61 5
3        83  -5
1       116   5
-1      119    -2
-3     1010     0
-5    811      3
-7   510       -5
-9  38        5
-11 15         -4
-13 3          3
-151           -1
Integral Khovanov Homology

(db, data source)

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